Method and apparatus for identifying propagation for routes with diagonal edges

ABSTRACT

Some embodiments provide a hierarchical method of routing nets within a particular region of a circuit layout. Each net has a set of pins. The method initially partitions the particular region into a first set of sub-regions. For each net, the method identifies a first route that connects a group of first-set sub-regions containing the first net&#39;s pins; where some of the routes have at least one route-edge that is at least partially diagonal. The method then partitions the sub-regions into a second set of smaller sub-regions. For a first net, the method identifies a propagation of the first-net&#39;s first route into the second-set sub-regions based on congestion between the second-set sub-regions. It then adjusts the congestion between the second set sub-regions based on the identified propagation. For a second net, the method then identifies a propagation of the second-net&#39;s first route into the second-set sub-regions based on congestion between the second-set sub-regions.

CLAIM OF BENEFIT TO PRIOR APPLICATION

[0001] This patent application claims the benefit of the earlier-filedU.S. Provisional Patent Application entitled “Routing Method andApparatus”, having serial No. 60/314,580, and filed Aug. 23, 2000; andU.S. Provisional Patent Application entitled “Routing Method andApparatus”, having serial No. 60/337,504, and filed Dec. 6, 2001.

FIELD OF THE INVENTION

[0002] The invention is directed towards method and apparatus foridentifying propagations for routes with diagonal edges.

BACKGROUND OF THE INVENTION

[0003] An integrated circuit (“IC”) is a device that includes manyelectronic components (e.g., transistors, resistors, diodes, etc.).These components are often interconnected to form multiple circuitcomponents (e.g., gates, cells, memory units, arithmetic units,controllers, decoders, etc.) on the IC. The electronic and circuitcomponents of IC's are jointly referred to below as “components.”

[0004] An IC also includes multiple layers of wiring (“wiring layers”)that interconnect its electronic and circuit components. For instance,many IC's are currently fabricated with metal or polysilicon wiringlayers (collectively referred to below as “metal layers”) thatinterconnect its electronic and circuit components. One commonfabrication model uses five metal layers. In theory, the wiring on themetal layers can be all-angle wiring (i.e., the wiring can be in anyarbitrary direction). Such all-angle wiring is commonly referred to asEuclidean wiring. In practice, however, each metal layer typically has apreferred wiring direction, and the preferred direction alternatesbetween successive metal layers. Many IC's use the Manhattan wiringmodel, which specifies alternating layers of preferred-directionhorizontal and vertical wiring. In this wiring model, the majority ofthe wires can only make 90° turns. However, occasional diagonal jogs aresometimes allowed on the preferred horizontal and vertical layers.

[0005] Design engineers design IC's by transforming circuit descriptionof the IC's into geometric descriptions, called layouts. To createlayouts, design engineers typically use electronic design automation(“EDA”) applications. These applications provide sets of computer-basedtools for creating, editing, and analyzing IC design layouts.

[0006] EDA applications create layouts by using geometric shapes thatrepresent different materials and devices on IC's. For instance, EDAtools commonly use rectangular lines to represent the wire segments thatinterconnect the IC components. These tools also represent electronicand circuit IC components as geometric objects with varying shapes andsizes. For the sake of simplifying the discussion, these geometricobjects are shown as rectangular blocks in this document.

[0007] Also, in this document, the phrase “circuit module” refers to thegeometric representation of an electronic or circuit IC component by anEDA application. EDA applications typically illustrate circuit moduleswith pins on their sides. These pins connect to the interconnect lines.

[0008] A net is typically defined as a collection of pins that need tobe electrically connected. A list of all or some of the nets in a layoutis referred to as a net list. In other words, a net list specifies agroup of nets, which, in turn, specify the interconnections between aset of pins.

[0009]FIG. 1 illustrates an example of an IC layout 100. This layoutincludes five circuit modules 105, 110, 115, 120, and 125 with pins130-160. Four interconnect lines 165-180 connect these modules throughtheir pins. In addition, three nets specify the interconnection betweenthe pins. Specifically, pins 135, 145, and 160 define a three-pin net,while pins 130 and 155, and pins 140 and 150 respectively define twotwo-pin nets. As shown in FIG. 1, a circuit module (such as 105) canhave multiple pins on multiple nets.

[0010] The IC design process entails various operations. Some of thephysical-design operations that EDA applications commonly perform toobtain the IC layouts are: (1) circuit partitioning, which partitions acircuit if the circuit is too large for a single chip; (2) floorplanning, which finds the alignment and relative orientation of thecircuit modules; (3) placement, which determines more precisely thepositions of the circuit modules; (4) routing, which completes theinterconnects between the circuit modules; (5) compaction, whichcompresses the layout to decrease the total IC area; and (6)verification, which checks the layout to ensure that it meets design andfunctional requirements.

[0011] Routing is a key operation in the physical design cycle. It isgenerally divided into two phases: global routing and detailed routing.For each net, global routing generates a “loose” route (also called pathor routing areas) for the interconnect lines that are to connect thepins of the net. The “looseness” of a global route depends on theparticular global router used. After global routes have been created,the detailed routing creates specific individual routing paths for eachnet.

[0012] While some commercial global routers today might allow anoccasional diagonal jog, these routers do not typically explore diagonalrouting paths consistently when they are specifying the routinggeometries of the interconnect lines. This, in turn, increases the totalwirelength (i.e., total length of interconnect lines) needed to connectthe nets in the layout. Therefore, there is a need for routing methodand apparatus that considers diagonal routing paths.

SUMMARY OF THE INVENTION

[0013] Some embodiments provide a hierarchical method of routing netswithin a particular region of a circuit layout. Each net has a set ofpins. The method initially partitions the particular region into a firstset of sub-regions. For each net, the method identifies a first routethat connects a group of first-set sub-regions containing the firstnet's pins; where some of the routes have at least one route-edge thatis at least partially diagonal. The method then partitions thesub-regions into a second set of smaller sub-regions. For a first net,the method identifies a propagation of the first-net's first route intothe second-set sub-regions based on congestion between the second-setsub-regions. It then adjusts the congestion between the second setsub-regions based on the identified propagation. For a second net, themethod then identifies a propagation of the second-net's first routeinto the second-set sub-regions based on congestion between thesecond-set sub-regions.

BRIEF DESCRIPTION OF THE DRAWINGS

[0014] The novel features of the invention are set forth in the appendedclaims. However, for purpose of explanation, several embodiments of theinvention are set forth in the following figures.

[0015]FIG. 1 illustrates an example of an IC layout.

[0016]FIG. 2 illustrates an IC layout that utilizes horizontal,vertical, and 45° diagonal interconnect lines.

[0017]FIG. 3 illustrates one manner of implementing an octagonal wiringmodel.

[0018]FIG. 4 presents a conceptual illustration of a recursive routingprocess performed by some embodiments of the invention.

[0019]FIG. 5 illustrates a design region of an IC layout that has beendivided into sixteen sub-regions.

[0020] FIGS. 6-8 illustrate three Steiner trees for a net illustrated inFIG. 5.

[0021]FIG. 9 illustrates two congestion grids.

[0022]FIG. 10 illustrates edges defined by the congestion grids of FIG.9.

[0023]FIG. 11 shows the diagonal edges of FIG. 10 slightly smaller.

[0024]FIG. 12 illustrates wiring paths across the edges of FIG. 10.

[0025]FIG. 13 illustrates a partitioning grid used by some embodiments.

[0026]FIG. 14 illustrates Manhattan and diagonal paths defined acrossedges created by the grid of FIG. 13.

[0027]FIG. 15 illustrates a length grid that decomposes eachcongestion-grid child slots of the grid of FIG. 13 into 4 slots, whileFIG. 16 illustrates a length grid that decomposes each of these childslots into 16 slots.

[0028]FIG. 17 illustrates that the partitioning of FIG. 15 creates 6paths for routing between the resulting 4 slots in each congestion-graphchild slot, while FIG. 18 illustrates that the partitioning of FIG. 16creates 42 paths for routing between the resulting 16 slots of eachcongestion-graph child slot.

[0029]FIG. 19 illustrates a process for adaptively selecting the wiringmodel, as well as the congestion and/or partitioning grids.

[0030]FIGS. 20 and 21 illustrate how some embodiments calculate thelength of an interconnect line connecting two nodes of a tree.

[0031]FIG. 22 illustrates a process that constructs one or more optimalSteiner trees for each possible net configuration with respect to apartitioning grid, and stores the trees and their attributes.

[0032]FIG. 23 pictorially illustrates sixteen tree nodes for sixteenslots created by a 4-by-4 partitioning grid.

[0033]FIG. 24 illustrates a process for identifying potential Steinernodes.

[0034]FIGS. 25A and 25B illustrate a process for that constructing oneor more minimum spanning trees (MST's) and computing each MST's lengthfor node configurations with two or more nodes.

[0035]FIG. 26 illustrates a process that calculates the routing-pathinformation and the path-usage probabilities.

[0036]FIGS. 27 and 28 respectively illustrate examples of path-usagecounts and path-usage probabilities for the Steiner trees of FIGS. 6-8.

[0037]FIG. 29 illustrates a compression technique for storingSteiner-tree routes for sets of net configurations.

[0038]FIGS. 30 and 31 illustrate one technique for grouping nodeconfigurations.

[0039]FIG. 32 illustrates a binary-search tree (“BST”) used for sortingstored trees.

[0040]FIG. 33 illustrates a process used to traverse the BST todetermine whether a tree was previously stored in the storage structure.

[0041]FIG. 34 illustrates a process that pre-tabulates routes and routeattributes for multiple wiring models.

[0042]FIG. 35A and FIG. 35B illustrate examples of closed and open nodeconfigurations.

[0043]FIG. 36 illustrates a process that pre-tabulates minimum closedtrees.

[0044]FIG. 37 illustrates a process that, for an open nodeconfiguration, pre-tabulates related closed node configurations that donot have antenna nodes.

[0045]FIG. 38 illustrates a process that identifies one or moreSteiner-tree routes for a net when the routes and closed nodeconfigurations are pre-tabulated according to the processes of FIGS. 36and 37.

[0046]FIG. 39 illustrates the software architecture of a router of someembodiments of the invention.

[0047]FIG. 40 illustrates a design region that is recursively dividedinto sets of 16 sub-regions.

[0048]FIG. 41 illustrates the data structure for a net list.

[0049]FIG. 42 illustrates a dbNet data structure.

[0050]FIG. 43 illustrates a simple pin data structure.

[0051]FIG. 44 illustrates a path data structure.

[0052]FIG. 45 illustrates a slot-net data structure.

[0053]FIG. 46 presents a graph that conceptually illustrates thehierarchy of slots defined by the router.

[0054]FIG. 47 presents a slot data structure.

[0055]FIG. 48 illustrates a circuit module data structure.

[0056] FIGS. 49-51 illustrate a process that is performed by aninitializer of the router of FIG. 39.

[0057]FIG. 52 illustrates a process performed by a slot manager of therouter of FIG. 39.

[0058]FIG. 53 illustrates a process performed by a solver of the routerof FIG. 39.

[0059]FIGS. 54 and 55 illustrate one manner for predicting thecongestion of the paths.

[0060]FIG. 56 illustrates a process for identifying routes for each netconfiguration and generating detour possibilities by adding fake pins tothe net configurations.

[0061]FIGS. 57 and 58 provide examples of how sub-optimal detour routesare generated by adding one or two fake pin configurations.

[0062]FIG. 59 illustrates another technique for identifying additionalroutes for a net configuration.

[0063]FIG. 60 illustrates a process that identifies additional routesfor a net configuration.

[0064]FIG. 61 illustrates one way for propagating a horizontal orvertical path between the current slot's child slots down into the slotsof the child slots.

[0065]FIGS. 62 and 63 illustrate two different ways for modeling thepropagation of a 45° diagonal path into the lower level child slots.

[0066]FIG. 64 illustrates a process for calculating the cost of eachroute in terms of three component costs.

[0067]FIG. 65 illustrates one example of a propagation possibility of apath.

[0068]FIGS. 66 and 67 present two examples that conceptually illustrateone manner of counting the number of vias.

[0069] FIGS. 68-70 illustrate three processes that work together tocompute the number of vias in a route.

[0070]FIGS. 71 and 72 illustrate the need for sharing constraints at theGcell level.

[0071]FIG. 73 illustrates a diagonal pair constraint.

[0072]FIG. 74 illustrates a mixed triplet constraint.

[0073]FIG. 75 illustrates a diagonal triplet constraint.

[0074]FIG. 76 illustrates a process that an ILP propagator performs insome embodiments.

[0075]FIGS. 77 and 78 illustrate one manner of estimating theavailability of the propagations.

[0076]FIGS. 79 and 80 illustrate one manner of enumerating and costingthe propagations.

[0077]FIG. 81 illustrates a process for performing follow-up propagationfor the current slot when the current slot is below the top-level slotbut above the leaf-level slot.

[0078]FIG. 82 illustrates a path from a follow-up path list that ispropagated.

[0079]FIG. 83 illustrates one a sequential-propagation process that isused in some embodiments.

[0080]FIG. 84 presents a computer system used to implement someembodiments of the invention.

DETAILED DESCRIPTION OF THE INVENTION

[0081] The invention provides method and apparatus for identifyingroutes for nets. In the following description, numerous details are setforth for purpose of explanation. However, one of ordinary skill in theart will realize that the invention may be practiced without the use ofthese specific details. In other instances, well-known structures anddevices are shown in block diagram form in order not to obscure thedescription of the invention with unnecessary detail.

[0082] Several embodiments of the invention's routing method andapparatus are described below. However, before discussing theseembodiments, several diagonal wiring architectures that can be used withthese embodiments are described in Section I.

I. Diagonal Wiring Architecture

[0083] Different embodiments of the invention can be used with differentwiring models. For instance, some embodiments are used with wiringmodels that include diagonal, horizontal, and vertical interconnectwires. In the discussion below, interconnect wires are also referred toas interconnects or interconnect lines. Also, as used in this document,an interconnect line is “diagonal” if it forms an angle other than zeroor ninety degrees with respect to the layout boundary. On the otherhand, an interconnect line is “horizontal” or “vertical” if it forms anangle of 0° or 90° with respect to one of the sides of the layout (e.g.,forms an angle of 0° or 90° with respect to the width of the layout).

[0084]FIG. 2 illustrates an IC layout 200 that utilizes horizontal,vertical, and45° diagonal interconnect lines. In this figure, thehorizontal lines 205 are the lines that are parallel (i.e., are at 0°)to the x-axis, which is defined to be parallel to the width 210 of thelayout. The vertical lines 215 are parallel to the y-axis, which isdefined to be parallel to the height 220 of the layout. In other words,the vertical interconnect lines 215 are perpendicular (i.e., are at90°)to the width of the IC layout. In addition, one set 225 of diagonallines are at +45° with respect to the width of the IC layout, whileanother set 230 are at −45° with respect to the width of the IC layout.In this document, the phrase “octagonal wiring model” is used to referto a wiring model that includes horizontal, vertical, and 45° diagonalinterconnect lines.

[0085]FIG. 3 illustrates one manner of implementing the octagonal wiringmodel. The wire model illustrated in this figure uses the notion ofhaving one preferred-wiring direction per layer. Specifically, FIG. 3illustrates five wire layers with each layer having its own preferreddirection. The first three layers 305-315 are Manhattan layers. In otherwords, the preferred direction for the interconnect lines in theselayers is either the horizontal direction or the vertical direction. Thepreferred wiring direction in these three layers typically alternates sothat no two consecutive layers have the same preferred wiring direction.However, in some cases, the wiring in consecutive layers is in the samedirection.

[0086] The next two layers 320 and 325 are diagonal layers. Thepreferred directions for the interconnect lines in the diagonal layersare ±45°. Also, as in the first three layers, the wiring directions inthe fourth and fifth layer are typically orthogonal (i.e., one layer is+45° and the other is −45°), although they do not have to be.

[0087] Several embodiments are described below with reference to theoctagonal wiring model illustrated in FIG. 3. However, one of ordinaryskill will understand that the invention can be used with any wiringmodel. For instance, the invention can be used with wiring architecturesthat are strictly diagonal (i.e., wiring architectures that do not havehorizontal and vertical direction wiring).

[0088] Also, some embodiments are used with non-45° diagonal wiring. Forexample, some embodiments are used with wiring models that utilizehorizontal, vertical and ±120° diagonal interconnect lines. In addition,some embodiments are used with wire models that do not specify apreferred direction for some or all the wire layers. For instance, someembodiments use an octagonal wiring model that allows horizontal,vertical, and 45° lines to exist on all wire layers.

II. Conceptual Flow

[0089]FIG. 4 presents a conceptual illustration of a recursive routingprocess performed by some embodiments of the invention. This routingprocess hierarchically defines routes for nets within a design region(also called a slot) of an IC layout. This region can be the entire IClayout, or a portion of this layout. Likewise, the IC layout can be alayout for the entire integrated-circuit chip or chips, or it can be alayout for a block (i.e., a portion) of an integrated-circuit chip.

[0090] The process initially defines (at 405) a partitioning grid thatdivides the IC region into several sub-regions. In the discussion below,the partitioned region is also referred to as the current slot, and thesub-regions resulting from the partitioning are also referred to as thecurrent slot's child slots.

[0091] In some embodiments, the partitioning grid is formed byintersecting cut lines. In some of these embodiments, the intersectingpartitioning lines are N horizontal and M vertical lines that divide theIC region into (N+1)(M+1) sub-regions, where N and M can equal anyinteger. For instance, these horizontal and vertical lines divide thereceived IC region into (1) four child slots when N and M equal 1, (2)nine child slots when N and M equal 2, (3) sixteen child slots when Nand M equal 3, or (4) twenty child slots when either N or M equals 4 andthe other equals 5.

[0092]FIG. 5 illustrates a design region 500 that has been divided intosixteen sub-regions (i.e., into child slots 0-15) by sets of threehorizontal and vertical partitioning lines. This figure also shows a net505 that includes five circuit modules 510, 515, 520, 525, and 530,which fall into four of the sixteen sub-regions. These four sub-regionsare slots 0, 1, 7, and 8.

[0093] Each net within the partitioned region (ie., within the currentslot) has one or more actual or virtual pins in the sub-regions definedby the partitioning grid. A net's actual pins are pins of circuitmodules in the design region, whereas the net's virtual pins areartificial pins that are set to account for the propagation of higherlevel routes into lower level child slots, as further described below.For each net, the set of sub-regions that contain that net's actual orvirtual pins represents the net's configuration with respect to thepartitioning grid.

[0094] For each particular net within the partitioned region, theprocess 400 uses (at 410) the net's configuration to identify one ormore routes (also called routing graphs or connection graphs) for thenet. Each route of a net provides a set of interconnect lines thatconnects the child slots (i.e., the sub-regions) that contain the net'spins.

[0095] To model each net's configuration with respect to the grid, eachchild slot that contains one or more of the net's pins is treated as anode (also called a vertex or point) of the routing graph. The nodes ofthe graph are then connected by edges (also called lines). According tosome embodiments, the routing graph can have edges that are completelyor partially diagonal.

[0096] Different embodiments use different types of graphs to define theinterconnect routes. In the embodiments described below, trees (e.g.,Steiner trees) are used as the routing graphs that connect the childslots containing the related net pins. FIGS. 6-8 illustrate threeoptimal Steiner trees 605, 705, and 805 for the net 505 in FIG. 5. TheseSteiner trees all have the same length. One of these trees (605) has aSteiner node (620). In addition, each of these trees has at least oneedge that is at least partially diagonal. In these examples, the routeruses the octagonal wiring model and therefore the diagonal edges are at45° degrees with respect to the layout boundary.

[0097] Before process 400 starts, some embodiments pre-compute and storeroutes for different configuration of child slots in a data storage. Atrun-time, the router in these embodiments identifies (at 410) some orall of the routes for a net by (1) identifying the configuration of eachnet with respect to the partitioning grid, and (2) retrieving from thedata storage the routes for the identified-configurations. Such anapproach frees the router from having to construct in real-time routesfor each net configuration. One such approach is described below inSection V.

[0098] Other embodiments, on the other hand, use net configurations togenerate routes in run time. Yet other embodiments use netconfigurations to retrieve and generate routes. For instance, someembodiments use net configurations to retrieve pre-tabulated routes forcertain nets while generating routes for other nets. One such approachis described below in Section V.

[0099] In some embodiments, the pre-tabulated or generated routes areoptimal routes. Some of these embodiments also have the router identifysub-optimal routes for each net configuration in the layout, in order toincrease the number of possible solutions for each net. One suchapproach is described below in Section VI.

[0100] For each net, the process 400 selects (at 415) one of the routesidentified for the net as the net's route at the current recursionlevel. The process selects the routes that optimize certain objectives,such as reducing wirelength and congestion. When the current slot'schild slots are to be partitioned to define smaller child slots, theprocess 400 then determines (at 420) the propagation of the selectedroutes into the smaller child slots. At this stage, the process mightalso add virtual pins to certain nets to account for such propagation.

[0101] Finally, when the child slots defined at 405 are not the slotsresulting from the last recursion operation, the process 400 recursivelyrepeats for each child slot defined at 405. By recursively repeating foreach defined child slot, the process 400 defines more and more detailedroutes for the nets in the current region. In other words, thisrecursive process 400 defines the routes in a hierarchical manner, wherethe process defines more detailed routes as the levels of the recursionhierarchy increase.

[0102] Some embodiments use different shaped partitioning grids fordifferent levels in the recursion process. The embodiments describedbelow, however, use same shaped partitioning grids for all the recursionlevels. At each recursion level, these embodiments simply adjust thecoordinates of the partitioning grid to match the coordinates of the ICregion at that recursion level. Using the same shaped partitioning gridsfor all the recursion levels has several advantages. For instance, theprocess can re-use the same set of pre-tabulated information for alllevels of the recursion process.

III. Multiple Grids

[0103] Some embodiments use one or more grids in addition to thepartitioning grid.

[0104] A. Multiple Congestion Grids.

[0105] Some embodiments use multiple congestion grids as the conceptualmodel for quantifying the capacity, and measuring the congestion, ofrouting paths between the sub-regions that are defined by thepartitioning grid. FIG. 9 illustrates two such congestion grids. Someembodiments described below use these two grids in conjunction with theoctagonal wiring model illustrated in FIG. 3.

[0106] In FIG. 9, the two grids are: (1) grid 905, which is formed by 3horizontal and 3 vertical lines, and (2) grid 910, which is formed byseven +45° diagonal lines and seven −45° diagonal lines. Grid 905 isused to specify the capacity and measure the congestion of horizontaland vertical routing paths, while grid 910 is used to specify thecapacity and measure the congestion of diagonal routing paths.

[0107] Specifically, as shown in FIG. 10, the grid 905 defines 12vertical edges (E0-E11) and 12 horizontal edges (E12-E23), while thegrid 910 defines 9−45° edges (E24, E26, E28, E30, E32, E34, E36, E38,E40) and 9 +45° edges (E25, E35, E37, E39, E41). In FIG. 10, thediagonal edges are shown to have endpoints, in order to simplify theidentification of these edges as they abut each other.

[0108] As shown in FIG. 10, each diagonal edge traverses the distancebetween the centers of two sub-regions that are defined by the firstgrid and that are at diagonally adjacent positions with respect to eachother. In other words, each diagonal edge connects the centers of twosub-regions that are aligned diagonally such that they abut at only oneof their corners. FIG. 11 shows the diagonal edges slightly smaller, inorder to simplify the appearance of these edges.

[0109] In some embodiments, grids 905 and 910 are also used to definerouting paths between the child slots of a partitioned region.Specifically, orthogonal to each edge defined by grids 905 and 910 is arouting path that can be used by a routing tree to connect the abuttingslots (i.e., the abutting sub-regions). For instance, FIG. 12illustrates 42 wiring paths across the 42 edges of FIG. 10. Horizontalpaths P0-P11 are defined across vertical edges E0-E11, vertical pathsP12-P23 are defined across horizontal edges E12-E23 that verticalrouting paths will intersect, +45° paths P24, P26, P28, P30, P32, P34,P36, P38, P40 are defined across −45° edges E24, E26, E28, E30, E32,E34, E36, E38, E40, and −45° paths P25, P27, P29, P31, P33, P35, P37,P41 are defined across +45° edges E25, E27, E29, E31, E33, E35, E37,E39, E41.

[0110] The congestion problem can be expressed and analyzed in terms ofeither the edge capacities or the path capacities, as these two sets ofcapacities are intertwined. The processes described below analyze thecapacity issue in terms of the path capacities. One of ordinary skillwill realize, however, that analogous processes can be used to analyzethe capacity issue in terms of edge capacities.

[0111] As further described below, some embodiments derive the capacityof each path from the size of the edge that the path intersects. Forinstance, some embodiments calculate the capacity of each particularpath by dividing the size of the corresponding orthogonal edge (i.e.,the size of the edge orthogonal to the particular path) with the pitchof metal layer corresponding to the particular path. Some embodimentsdefine the pitch of a metal layer as the line-to-via pitch. Someembodiments define the line-to-via pitch as the minimum requireddistance between interconnect lines on that metal layer, plus ½ thewidth of the line, plus ½ the width of the via including the metaloverlap.

[0112] In some embodiments, the capacities of the diagonal paths differfrom the capacities of the Manhattan paths. This can be due to thediffering size of the edges that are orthogonal to the diagonal andManhattan paths. It can also be due to the pitch of the diagonal linesbeing different from the pitch of the Manhattan lines. It can further bedue to the pitch of one layer being different than the pitch of anotherlayer. For example, in some embodiments, the capacities of the −45°diagonal paths differ from the capacities of the 45° diagonal paths,when the pitch of the −45° metal layer differs from the pitch of the 45°metal layer.

[0113] In FIG. 9, the grid 905 is the same as the partitioning gridillustrated in FIG. 5. However, one of ordinary skill will appreciatethat both congestion grids can differ from the partitioning grid. Inaddition, even though FIG. 9 illustrates two congestion grids 905 and910 for some embodiments, one of ordinary skill will appreciate thatother multi-grid arrangements can be used by other embodiments.

[0114] Some embodiments generally define the number and structure ofcongestion grids based on the number of wiring directions and wiringlevels of the wiring model used to design the design layout and/or theIC. For instance, some embodiments use a wiring model that allowsrouting in horizontal, vertical, +120° diagonal, and −120° diagonaldirections. For such a wiring model, two congestion grids can be used.Like grid 905, the first grid could be formed by intersecting horizontaland vertical lines, in order to define the capacity and measure thecongestion of vertical and horizontal routing paths. The second gridcould be used to define the capacity and measure the congestion of ±120°diagonal routing paths. This second grid could be similar to the firstgrid, except that the axis of the second grid would be rotated 120° withrespect to the axis of the first grid. In other words, this second gridcould be formed by a number of intersecting ±30° lines.

[0115] B. Congestion and Length Grids.

[0116] Some embodiments use (1) a first grid to partition an IC regionand measure congestion in this region, and (2) a second grid to measurewirelength costs in the region. FIGS. 13-18 illustrate several suchembodiments. These embodiments use the wiring model that includeshorizontal, vertical, and +45° interconnect lines. One of ordinary skillwill understand that other embodiments use other wiring models (such asones that use ±120° lines).

[0117]FIG. 13 illustrates a first grid 1305 that some embodiments use topartition an IC region into 16 sub-regions. This grid also defines 24edges E0-E23 that are used to measure congestion of Manhattan andnon-Manhattan interconnect lines in the region. As shown in FIG. 14, 24Manhattan paths P0-P23 and 48 diagonal paths P24-P71 are defined acrossthese 24 edges E0-E23. Each path represents one or more tracks of wiringin the path's direction across the path's corresponding edge.

[0118] Vertical edges E0-E11 are used to measure congestion of wires(i.e., of interconnect lines) that cross these vertical edges in thedirection of horizontal paths P0-P11 and ±45° diagonal paths P24-P29,P38-P43, P52-P57, and P66-P71. Similar horizontal edges E12-E23 are usedto measure congestion of wires that cross these horizontal edges in thedirection of vertical paths P12-P23 and ±45° diagonal paths P30-P37,P44-P51, and P58-P65.

[0119] Some embodiments define each route in terms of the paths P0-P71.Paths P0-P71 are also used to measure congestion in the IC region. Insome embodiments, the capacity along the diagonal paths P24-P71 is lessthan the capacity along the Manhattan paths P0-P23. For instance, someembodiments specify that (1) each Manhattan path represents 8-tracks ofwires at the lowest-level child slot (i.e., at the Gcell level), and (2)each diagonal path represents 5-tracks of wires at the Gcell level whenthe diagonal and Manhattan layers have the same pitch. Some embodimentsspecify less than 5-tracks for a diagonal path at the Gcell level, whenthe pitch of the diagonal path's layer is greater than the pitch of aManhattan path layer.

[0120] As mentioned above, some embodiments use a second grid to measurethe wirelength costs of the routes in the region. This length griddecomposes each congestion-grid child slot into smaller slots. Forinstance, FIG. 15 illustrates a length grid that decomposes each of the16 congestion-grid child slots of grid 1305 into 4 slots, while FIG. 16illustrates a length grid that decomposes each of the 16 congestion-gridchild slots of grid 1305 into 16 slots. Other embodiments use othertypes of grids (e.g., a 3×5 grid) to decompose the congestion-grid childslots.

[0121]FIG. 17 illustrates that the 2×2 partitioning of FIG. 15 define 6paths for routing between the resulting 4 slots in each congestion-graphchild slot, while FIG. 18 illustrates that the 4×4 partitioning of FIG.16 defines 42 paths for routing between the resulting 16 slots of eachcongestion-graph child slot. In addition, each type of partitioningdefines several paths between the length-grid slots of adjacentcongestion-grid child slots. These paths will be further describedbelow.

[0122] The length grid can be used to estimate the wirelength cost ofeach net's route by identifying one or more segments that traverse thelength-grid paths to connect all of the net's pins. In other words, fora net's route, an estimated wirelength cost is the length of a set oflength-grid paths that (1) connect the length-grid slots that containthe net's pins, and (2) cross the same congestion-graph edges in thesame direction as the congestion-graph paths used by the net's route.The wirelength cost of a set of length-grid paths includes the cost ofthe interior paths (i.e., length-grid paths inside congestion-gridslots) that connect the length-grid slots containing the net's pins,plus the cost of the periphery length-grid path(s) that cross theincident congestion-graph edge(s).

[0123] In some embodiments, propagating a congestion-grid path to aperiphery length-grid path (i.e., a length-grid path between congestiongrid slots) might require the setting of a virtual pin in the net's pinconfiguration with respect to the length grid. Accordingly, the interiorlength-grid paths connect the length-grid slots that contain actual orvirtual pins of the net. Also, as mentioned above, the peripherylength-grid paths (i.e., the length-grid paths across congestion-graphedges) cross the congestion-graph edges in the same direction as thecongestion-graph paths used by the net's route.

[0124]FIGS. 17 and 18 illustrate diagonal length-grid paths between thelength-grid slots of diagonally-adjacent congestion-grid child slots. Inthese figures, such diagonal length-grid paths are circled with dashedlines. Some embodiments define such diagonal length-grid paths whileothers do not.

[0125] Also, some of the embodiments that have diagonal length-gridpaths between diagonally-adjacent congestion-grid child slots use aparticular convention to correlate these diagonal length-grid paths withthe diagonal congestion-graph paths. In some embodiments, such −45°length-grid paths are tied either to their corresponding bottom and leftcongestion-graph paths or to their corresponding top and rightcongestion-graph paths. For instance, when a −45° length grid path isused between congestion-graph child slots 9 and 12, some embodimentsincrement the path-usage of paths 53 and 59 by one, while otherembodiments increment the path-usage of paths 61 and 67 by one. (Paths53, 59, 61, and 67 are illustrated in FIG. 14.)

[0126] Analogously, some embodiments tie +45° length-grid paths eitherto their corresponding bottom and right congestion-graph paths or totheir corresponding top and left congestion-graph paths. For instance,when a +45° length grid path is used between congestion-graph childslots 8 and 13, some embodiments increment the path-usage of paths 58and 66 by one, while other embodiments increment the path-usage of paths52 and 60 by one. (Paths 52, 58, 60, and 66 are illustrated in FIG. 14.)

[0127] Alternatively, some embodiments tie diagonal length-grid pathsbetween diagonally-adjacent congestion-grid child slots to only one ofthe four surrounding congestion-graph paths, and assign one additionaltrack to the capacity of this congestion-graph path. Some embodimentstie a −45° length-grid path to its corresponding left congestion-graphpath, and tie +45° length-grid path to its corresponding rightcongestion-graph path. Under this approach, some embodiments associatethe −45° length grid path between congestion-graph child slots 9 and 12with path 59, and assign a capacity of 6 to path 59 at the Gcell levelwhile assigning a capacity of 5 to path 53.

[0128] Yet other embodiments do not correlate the diagonal length-gridpaths between congestion-graph child slots with the diagonalcongestion-graph paths P0-P71. Instead, these embodiments define 18additional diagonal congestion-graph paths between the 18 pairs ofdiagonally-adjacent congestion graph slots. Each of these 18 additionalcongestion paths corresponds to a particular length-grid path. Also, atthe Gcell level, some embodiments define each of these 18 additionalpaths to be 1-track wide.

[0129] Different embodiments use the congestion and length gridsdifferently. For instance, some embodiments identify routes based on netconfigurations with respect to the congestion grid 1305, and then usethe length and congestion grids to compute wirelength and congestioncosts of the identified routes. Other embodiments successively expand aroute for a net through the length grid. For each expansion or potentialexpansion, these embodiments use the length grid to cost the expansionor potential expansion. If the expansion or potential expansion crossesone of the congestion grid edges, these embodiments factor a congestioncost for it. Also, as mentioned above, some embodiments only define eachroute eventually in terms of the paths P0-P71 defined across thecongestion grid, while others do not.

IV. Adaptive Selection of Wiring Model

[0130] Some embodiments adaptively select their wiring model based onthe aspect ratio (height-to-width ratio) of the design region (i.e., theregion being designed). FIG. 19 illustrates a process 1900 for makingsuch an adaptive selection. This process is typically performed beforedefining the partitioning grid at 405 of process 400. In someembodiments, the designer performs some or all operations of thisprocess manually, while in other embodiments the router performs some orall the operations of this process in an automated fashion.

[0131] This process initially identifies (at 1905) the aspect ratio ofthe design region. To identify the aspect ratio, the process cancalculate this ratio based on the dimensions of the design region, or itcan retrieve a pre-tabulated aspect ratio for the design block. Theprocess next selects (at 1910) a wiring model based on the identifiedaspect ratio. In some embodiments, the process 1900 then adaptivelyselects (at 1915) the partitioning and/or congestion grids. In someembodiments, the process adaptively selects the partitioning and/orcongestion grids based on the wiring model.

[0132] Adaptive selection of the wiring model allows a design region tobe routed with a view to achieving certain design objectives (e.g.,minimizing wirelength and congestion). For instance, when designing acircuit block that has a relatively large aspect ratio (i.e., a circuitblock that is tall and skinny), some embodiments adaptively select awiring model that allows routing in horizontal, vertical, ±120° diagonaldirections, because such a wiring model reduces wirelength andcongestion for routing such a circuit block. For such a wiring model,some embodiments use a first congestion grid (like grid 905 of FIG. 9)that is formed by intersecting horizontal and vertical lines, and asecond congestion grid that is formed by intersecting ±30° lines, asdescribed above.

[0133] Also, for such a wiring model and IC region, some embodiments usea partitioning grid that divides the IC region into smaller regions thathave large aspect ratio. In some such embodiments, diagonally-adjacentpartitioned regions have their centers offset by 120° from each other,so that their centers can be connected by 120° diagonal lines.

[0134] Numerous other wiring models can be used for a design block witha large aspect ratio. For instance, another wiring model would be onethat would allow routing in the horizontal, vertical, ±45° diagonal, and±120 diagonal directions. For such a wiring model, some embodimentsmight use the following three congestion grids: (1) a first grid that isfor horizontal and vertical paths and that is formed (like grid 905 ofFIG. 9) by intersecting horizontal and vertical lines, (2) a second gridthat is for ±120° paths and that is formed by intersecting ±30° lines,and (3) a third grid that is for ±45° paths and that is formed byintersecting ±45° diagonal lines.

[0135] Similarly, numerous wiring models can be used for a design blockwith a small aspect ratio (i.e., a block that is short and wide). Forinstance, some embodiments might adaptively select for such a block awiring model that allows routing in horizontal, vertical, ±30° diagonaldirections. For such a wiring model, some embodiments use the followingtwo congestion grids (1) a first grid that is for horizontal andvertical paths and that is formed (like grid 905 of FIG. 9) byintersecting horizontal and vertical lines, and (2) a second grid thatis for ±30° diagonal paths and that is formed by intersecting ±120°lines.

[0136] For such a wiring model and congestion grid, some embodiments usea partitioning grid that divides the IC region into smaller regions thathave small aspect ratio. In some such embodiments, diagonally-adjacentpartitioned regions have their centers offset by 30° from each other, sothat their centers can be connected by 30° diagonal lines.

[0137] Another wiring model for such a block would be one that wouldallow routing in the horizontal, vertical, ±45° diagonal, and ±30°diagonal directions. For such a wiring model, some embodiments use thefollowing three congestion grids: (1) a first grid that is forhorizontal and vertical paths and that is formed (like grid 905 of FIG.9) by intersecting horizontal and vertical lines, (2) a second grid thatis for ±30° diagonal paths and that is formed by intersecting ±120°lines, and (3) a third grid that is for ±45° diagonal paths and that isformed (like grid 910 of FIG. 9) by intersecting ±45° lines.

[0138] When the design block is square, some embodiments might select aperfectly symmetrical wiring model, such as the five-layer octagonalwiring model discussed above by reference to FIG. 3. However, in thissituation, other embodiments might select more complicated wiringmodels. For instance, some embodiments might select a nine-layer wiringmodel, which includes the first five layers illustrated in FIG. 3 plusanother four layers that are similar to layers 2-5 illustrated in FIG.3. One set of congestion grids for such a wiring model can include theabove-mentioned grids 905 and 910 for layers 2-5, and another two gridssimilar to grids 905 and 910 for layers 6-9.

[0139] Another complicated symmetrical wiring model that someembodiments might use is similar to the 9-layer model described above,except that the preferred directions in the last four layers (i.e.,layers 6-9) have been shifted by 22.5° in the same direction. This wouldresult in a wiring model that would provide 16-directions of routingfrom any given point, where each routing-path direction is 22.5° fromits neighboring routing-path directions. One set of congestion grids forsuch a wiring model can include grids 905 and 910 mentioned above forlayers 1-5, and another two grids that are 22.5°-shifted versions ofgrids 905 and 910 for layers 6-9.

V. Pre-tabulating Routing Information

[0140] As mentioned above, some embodiments pre-compute and store routesfor different configuration of child slots in a storage structure. Atrun-time, the router in these embodiments identifies some or all of theroutes for a net by (1) identifying the configuration of each net withrespect to the partitioning grid, and (2) retrieving from the storagestructure the routes for the identified-configurations. One manner ofpre-tabulating Steiner-tree routes is described below by reference toFIGS. 20-34.

[0141] Other embodiments, on the other hand, use net configurations togenerate routes in real time. Yet other embodiments use netconfigurations to retrieve and generate routes. For instance, someembodiments use net configurations to retrieve pre-tabulated routes forcertain nets and to generate routes for other nets. One such approach isdescribed below by reference to FIGS. 35-38.

[0142] A. Pre-tabulating Steiner-Tree Routes

[0143] FIGS. 20-34 illustrate one manner of pre-tabulating Steiner treesthat model possible net configurations with respect to the partitioninggrid. The pre-tabulation of attributes of these trees are also describedbelow. As mentioned above, a router can use such pre-tabulated routesand/or attributes during the routing process. Other EDA applications canalso use these routes and/or attributes. For instance, as disclosed inU.S. patent application entitled “Recursive Partitioning PlacementMethod and Apparatus”, filed on Dec. 6, 2000, and having the Ser. No.09/732,181, placers might use pre-tabulated wirelength, path-countvalues, and/or path-probability values to measure the cost of aplacement.

[0144] 1. Calculating the Length of an Interconnect Line Connecting TwoNodes of a Tree.

[0145]FIGS. 20 and 21 illustrate how some embodiments calculate thelength of an interconnect line connecting two nodes of a tree. Theseembodiments perform these operations by treating the two nodes asopposing corners of a bounding box that has a long side (L) and a shortside (S).

[0146] FIGS. 20 presents an example of a bounding-box 2005 for two nodes2035 and 2040. As shown in this figure, the line 2010 traverses theshortest distance between nodes 2035 and 2040 for layouts that utilizehorizontal, vertical, and diagonal interconnect lines. This line ispartially diagonal. Specifically, in this example, one segment 2020 ofthis line is diagonal, while another segment 2015 is horizontal.

[0147] Equation (A) below provides the distance traversed by line 2010(ie., the minimum distance between the nodes 2035 and 2040).

Distance=[L−{S(cos A/sin A)}]+S/sin A  (A)

[0148] In this equation, “L” is the box's long side, which in thisexample is the box's width 2025 along the x-axis, while “S” is the box'sshort side, which in this example is its height 2030 along the y-axis.Also, in this equation, “A” is the angle that the diagonal segment 2020makes with respect to the long side of the bounding box. In someembodiments, this angle A corresponds to the direction of some of thediagonal interconnect lines in the layout. For instance, in someembodiments, the angle A equals 45° when the layout uses the octagonalwiring model illustrated in FIG. 3.

[0149] Equations (B)-(D) below illustrate how Equation (A) is derived.The length of the line 2010 equals the sum of the lengths of its twosegments 2015 and 2020. Equation (B) provides the length of thehorizontal segment 2015, while Equation (C) provides the length of thediagonal segment 2020.

Length of 2015=L−(Length of 2020)*(cos A)  (B)

Length of 2020=S/sin A  (C)

[0150] Equations (B) and (C) can be combined to obtain Equation (D)below, which when simplified provides Equation (A) above.$\begin{matrix}\begin{matrix}{{Distance} = {{{Length}\quad {of}\quad 2015} + {{Length}\quad {of}\quad 2020}}} \\{= {L - {{S/\sin}\quad A*\left( {\cos \quad A} \right)} + {{S/\sin}\quad A}}}\end{matrix} & (D)\end{matrix}$

[0151] When the angle A equals45° , Equation (A) simplifies to Equation(E) below.

Distance=L+S*(sqrt(2)−1)  (E)

[0152] When the bounding box has no width or height, then the boundingbox is just a line, and the minimum distance between the opposingcorners of this line is provided by the box's long (and only) side,which will be a horizontal or vertical line. When the bounding box hasequal sized height and width (i.e., when it is a square) and the angle Ais 45°, a line that is completely diagonal specifies the shortestdistance between the box's two opposing corners.

[0153]FIG. 21 illustrates a process 2100 that identifies a bounding boxfor two nodes of a tree, and calculates the length of an interconnectline connecting the two nodes based on the bounding box's dimensions andEquation (A). This process initially (at 2105) determines whether thex-coordinate (X₁) of the first node is greater than the x-coordinate(X₂) of the second node. If so, the process defines (at 2110) thex-coordinate (X₁) of the first node as the maximum x-coordinate(X_(Max)), and the x-coordinate (X₂) of the second node as the minimumx-coordinate (X_(Min)). Otherwise, the process defines (at 2115) thex-coordinate (X₂) of the second node as the maximum x-coordinate(X_(Max)), and the x-coordinate (X₁) of the first node as the minimumx-coordinate (X_(Min)).

[0154] Next, the process determines (at 2120) whether the y-coordinate(Y₁) of the first node is greater than the y-coordinate (Y₂) of thesecond node. If so, the process defines (at 2125) the y-coordinate (Y₁)of the first node as the maximum y-coordinate (Y_(Max)), and they-coordinate (Y₂) of the second node as the minimum y-coordinate(Y_(Min)). Otherwise, the process defines (at 2130) the y-coordinate(Y₂) of the second node as the maximum y-coordinate (Y_(Max)), and they-coordinate (Y₁) of the first node as the minimum y-coordinate(Y_(Min)).

[0155] The process then defines (at 2135) the four coordinates of thebounding box as (X_(MIN), Y_(MIN)), (X_(MIN), Y_(MAX)), (X_(MAX),Y_(MIN)), and (X_(MAX), Y_(MAX)). Next, the process determines (at 2140)the bounding-box's width and height. The process determines (1) thewidth by taking the difference between the box's maximum and minimumx-coordinates, and (2) the height by taking the difference between thebox's maximum and minimum y-coordinates. The process then determines (at2145) whether the computed width is greater than the computed height. Ifso, the process defines (2150) the width as the long side and the heightas the short side. Otherwise, the process defines (at 2155) the width asthe short side and the height as the long side.

[0156] After 2150 or 2155, the process then uses (at 2160) theabove-described Equation (A) to compute the length of the shortestinterconnect line that connects the two nodes. The process then ends.

[0157]2. Constructing Steiner Trees for All Possible Net Configurationsand Pre-tabulating Length and Wiring Path Information for each Tree.

[0158]FIG. 22 illustrates a process 2200 that (1) constructs one or moreoptimal Steiner trees for each possible net configuration with respectto a partitioning grid, (2) stores the length of each constructedSteiner tree in a storage structure, such as a look-up table (“LUT”),(3) computes and stores the probability of the trees using each wirepath in the grid, and (4) stores the identity of each tree by storingthe wire paths for each tree in the storage structure.

[0159] This process 2200 is performed before the router starts itsoperation, so that the router does not have to construct in real-timeSteiner trees for each net configuration. Instead, because of process2200, the router needs only (1) to identify the configuration of eachnet with respect to the partitioning grid, and (2) to retrieve storedattributes for the identified configuration.

[0160] As shown in FIG. 22, process 2200 initially starts (at 2205) bydefining a tree node for each sub-region (also called slot) defined by aparticular partitioning grid. FIG. 23 pictorially illustrates sixteentree nodes 2305 for sixteen slots created by a 4-by-4 partitioning grid.These nodes represent all the potential nodes of trees that model theinterconnect topologies of all the net configurations. In FIG. 23, theidentified nodes are positioned at the center of each slot. In otherembodiments, the nodes can uniformly be defined at other locations inthe slots (e.g., can be uniformly positioned at one of the corners ofthe slots).

[0161] Next, the process 2200 defines (at 2210) a set N of possible nodeconfigurations. When the partitioning grid defines Y (e.g., four, nine,sixteen, twenty, etc.) sub-regions, set N includes 2^(Y) nodeconfigurations. After defining the set N of possible nodeconfigurations, the process 2200 select (at 2215) one of the possiblenode configurations NT from this set.

[0162] The process then constructs (at 2220) one or more minimumspanning trees (“MST's”) for the node configuration selected at 2215,and computes each constructed tree's length (MST_Cost). As furtherdescribed below, each constructed MST can have edges that are completelyor partially diagonal. A node configuration that has less than two nodesdoes not have a MST, and accordingly its MST_Cost is zero. In addition,FIG. 25A illustrates a process 2500 that constructs one or more MST'sand computes each MST's length for node configurations with two or morenodes. This process 2500 will be described further below.

[0163] After 2220, the process 2200 identifies (at 2225) potentialSteiner nodes, and then defines (at 2230) all possible sets of Steinernodes. One manner of identifying potential Steiner nodes will beexplained below by reference to FIG. 24. Each set of Steiner nodes thatis defined at 2230 includes one or more of the Steiner nodes identifiedat 2225. Also, each defined set of Steiner nodes has a maximum size thatis two nodes less than the number of nodes in the selected nodeconfiguration.

[0164] For each set of Steiner nodes identified at 2230, the processthen (at 2240) (1) constructs one or more MST's of the nodes in theselected node configuration and the selected Steiner-node set, and (2)computes and stores each MST's length (MST_Cost). Each constructed MSTcan use edges that are completely or partially diagonal. As mentionedabove, a node configuration that has less than two nodes does not have aMST, and accordingly its MST_Cost is zero. In addition, FIG. 25Aillustrates a process 2500 that constructs one or more MST's andcomputes each MST's length for node configurations with two or morenodes. This process 2500 will be described further below.

[0165] Next, the process 2200 selects (at 2240) the shortest set of theMST's generated at 2220 or 2235 as the optimal Steiner trees for thecurrent node configuration. In other embodiments, this process usesother criteria to select its set of Steiner trees. At 2240, the processalso stores in a storage structure (such as a LUT) the length (MST_Cost)of the Steiner tree or trees identified at 2240.

[0166] After selecting one or more Steiner trees for the current nodeconfiguration at 2240, the process 2200 calls (at 2245) a process 2600to calculate the routing-path information and the path-usageprobabilities resulting from the selected Steiner trees. This processwill be described below by reference to FIG. 26.

[0167] The process 2200 next determines (at 2250) whether it hasexamined all the node configurations in the set N defined at 2210. Ifnot, the process returns to 2215 to select an unexamined nodeconfiguration from this set and then repeat operations 2220-45 for thenewly selected node configuration. Otherwise, the process ends.

[0168]FIG. 24 illustrates a process 2400 for identifying potentialSteiner nodes. The process 2400 of FIG. 24 only needs to be performedfor node configurations with three or more nodes, because each set ofSteiner nodes defined at 2220 has a maximum size that is two nodes lessthan the number of nodes in the selected node configuration (i.e.,because Steiner-node sets are not defined at 2220 for nodeconfigurations with two or fewer nodes).

[0169] The process 2400 starts (at 2405) by initializing a set P ofpotential Steiner nodes equal to all the nodes defined at 2205 that arenot part of the node configuration selected at 2215. This process thenselects (at 2410) one of the potential Steiner nodes. Next, the process2400 determines (at 2415) whether the node (Q) selected at 2410 is on ashortest path between any two nodes in the selected node configuration.To make this determination, the process determines whether any two nodes(B and C) exit in the node configuration such that the distance betweenthe two nodes (B and C) equals the sum of (1) the distance between thefirst node (B) and the selected node (Q), and (2) the distance betweenthe second node (C) and the selected node (Q). In some embodiments, theprocess uses the above-described process 2100 and Equation (A) tocalculate the distance between any pair of nodes.

[0170] If the process determines that the node Q selected at 2410 lieson a shortest path between any two nodes in the node configuration, theprocess keeps (at 2420) the selected node in the set P of potentialSteiner nodes, flags this node as a node that it has examined, andtransitions to 2430, which is described below. On the other hand, if theselected node (Q) is not on the shortest path between any two nodes inthe selected node configuration, the process removes (at 2425) theselected node from the set P of potential Steiner nodes, and transitionsto 2430.

[0171] At 2430, the process determines whether it has examined all thenodes in the set of potential Steiner nodes. If not, the process returnsto 2410 to select another node in this set so that it can determine at2415 whether this node is on a shortest path between any two nodes inthe selected node configuration. When the process determines (at 2430)that it has examined all the nodes in the set of potential Steinernodes, it ends.

[0172]FIG. 25A illustrates a process 2500 that the process 2200 of FIG.22 uses at 2220 and 2235 to construct minimum spanning trees. A minimumspanning tree for a node configuration is a tree that has N−1 edges thatconnect (i.e., span) the N nodes of the configuration through theshortest route, which only branches (i.e., starts or ends) at the nodes.

[0173] In some embodiments of the invention, the edges of the MST's canbe horizontal, vertical, or diagonal. The diagonal edges can becompletely or partially diagonal. Also, when the layouts use diagonalinterconnect lines (e.g., ±45° interconnect lines), the diagonal edgesof the MST's can be in the same direction (e.g., can be in ±45°direction) as some of the diagonal interconnect lines in the layout. Forinstance, when the layout uses an octagonal wiring model (i.e., useshorizontal, vertical, and 45° diagonal lines), some embodimentsconstruct MST's that have horizontal, vertical, and 45° diagonal edges.

[0174] By treating the two nodes of each edge of an MST as two opposingcorners of a bounding box, the length of each edge can be obtained byusing the above-described process 2100 and Equation (A).

Distance=[L−{S(cos A/ sin A)}]+S/sin A  (A)

[0175] As described above, in this equation, “L” is the box's long side,“S” is the box's short side, and “A” is the angle that the diagonalsegment of the edge makes with respect to the long side of the boundingbox.

[0176] The process 2500 starts whenever the process 2200 calls it (at2220 or 2235) (1) to construct one or more MST's for a set M of nodes,and (2) to calculate the length of each constructed MST. This processinitially (at 2505) sets the MST length (MST_Cost) to zero. Next, theprocess (at 2510) (1) selects a node from the received set M of nodes asthe first node of the spanning tree, and (2) removes this node from thisset M.

[0177] The process 2500 then calls (at 2515) a process 2550 illustratedin FIG. 25B, in order to identify one or more linked set of nodes thatrepresent one or more complete MST's. The process 2550 is a recursiveprocess that, when called, receives (1) a set of nodes that representsan incomplete MST, and (2) a set of nodes M that are the nodes of thenode configuration that have not yet been added to the receivedincomplete MST. When the process 2500 calls the process 2550, itsupplies the process 2550 the first node selected at 2510, and themodified set of remaining nodes M. In response, the recursive process2550 returns one or more linked set of nodes that represent one or moreMST's, as further described below. The process 2550 might return morethan one copy of the same linked node set. Accordingly, after theprocess 2500 receives one or more linked node sets from process 2550,the process 2500 eliminates (at 2520) any duplicate copy of the samereceived linked node set, so that there is only one copy of eachreceived node set. After 2520, the process 2500 returns the constructedMST's and their lengths, and then ends.

[0178] As shown in FIG. 25B, the process 2550 defines (at 2525) aremainder set R of nodes equal to the set M of nodes that it receivedwhen it was called. At 2530, the process 2550 selects a node from theremaining node set R, and removes the selected node from the set ofremaining nodes. The process then computes and stores (at 2535) thedistance between the node selected at 2530 and each current node of thereceived incomplete MST. The distance between the selected node and eachnode can be traversed by an edge that is completely or partiallydiagonal. Hence, in some embodiments, the process 2550 uses theabove-described process 2100 and Equation (A) to compute the minimumdistance between the selected node and each node.

[0179] Next, the process determines (at 2540) whether there is any noderemaining in set R. If so, the process returns to 2530 to select anothernode from this set, so that it can compute (at 2535) the distancebetween this node and the current nodes of the spanning tree. Otherwise,the process (at 2545) identifies the smallest distance recorded at 2535,and identifies the node pair or pairs (where, in each pair one node isfrom the received set M and one node is from the received MST) thatresulted in this distance. The process 2550 then (at 2555) adds theidentified smallest distance to the MST length (MST_Cost). Next, theprocess determines (at 2560) whether it identified more than one pair ofclosest nodes. If not (i.e., if the identified minimum distance isbetween only one node in set M and only one node in the MST), theprocess (at 2565) (1) defines a tree node corresponding to the set-Mnode identified at 2545, (2) removes the identified node from set M, and(3) links the defined tree node to the MST node identified at 2545. At2565, the process 2550 also recursively calls itself and supplies themodified MST and the modified set M, when set M is empty after theremoval of the identified node. On the other hand, when the modified setM is empty, the process 2550 transitions from 2565 to 2575, where itreturns one set of nodes that represents one complete MST that wascompleted by the linking at 2565. After 2575, the process 2550terminates.

[0180] If the process 2550 determines (at 2560) that it identified (at2545) more than one “closest” node pairs, it sequentially andrecursively tries to obtain a complete MST based on each identifiedclosest node pair. In other words, this process initially selects one ofthe identified node pairs, and then (1) removes the selected pair'sset-M node from set M, (2) links the removed node to the pair's MSTnode, and (3) recursively repeats for the modified MST and the modifiedset M. Once the process 2550 receives the results of this recursion(i.e., when this process receives the complete MST's for the selectednode pair), it then selects the next identified node pair, and performsthe same three operations in order to obtain complete MST's based onthis node pair. The process continues in this manner until it generatesthe MST's based on each of the identified “closest” node pairs. Aftersequentially processing each identified node pair, the process returns(at 2575) the MST's, and then terminates.

[0181]FIG. 26 illustrates a process 2600 that calculates therouting-path information and the path-usage probabilities resulting fromthe Steiner trees selected at 2250. This process 2600 starts each timeprocess 2200 calls it at 2245 and provides it with a set of Steinertrees.

[0182] The process 2600 starts by initializing (at 2605) a global countvariable that stores a count value for each path. For each receivedtree, the process initializes (at 2610) a bit string for storing thattree's routing path information. The process then selects (at 2615) areceived Steiner tree, and selects (at 2620) one of the edges in thetree (i.e., selects a pair of linked nodes in the tree, where thesenodes were linked at 2565 or 2570 of process 2550). Next, the processdetermines (at 2625) whether more than one set of paths exist to routethe selected tree edge (i.e., to connect the selected pair of nodes). Insome embodiments, the process retrieves the path values for the selectedtree edge from a storage structure (e.g., a LUT) that stores path-usagevalues for any combination of the tree slot nodes. In other words, thisstorage structure maps the endpoints of each possible tree edge withinthe grid to a set of path-usage values.

[0183] When the tree edge endpoints are not adjacent (i.e., when thepair of nodes selected at 2620 are not adjacent), more than one optimalroute might exist between the endpoints (i.e., between the node pairs).Hence, the path-usage values in the LUT might specify values formultiple optimal routes.

[0184] Two sets of node connections that could represent the threeSteiner trees shown in FIGS. 6-8 are (1)node set formed by node 610-node615-Steiner node 620-node 625-node 630 for the Steiner tree 605 of FIG.6, and (2) the node set formed by node 625-node 610-node 615-node 635for the Steiner trees of FIGS. 7 or 8.

[0185] In the first set of nodes representing the Steiner tree 605 ofFIG. 6, only one route exists between any two connected pairs of nodes.Hence, for any pair from this set, the mapping LUT would return 42values, with all the values equal to 0 except the value for the pathbetween the selected node pair. This non-zero value would be 1 toindicate that only one route exists between the selected node pair.

[0186] On the other hand, for the second set of nodes representingeither Steiner tree 705 or 805, two routes exist between nodes 615 and630. The Steiner tree 705 uses one of these routes, while the Steinertree 805 uses the other. For this node pair (ie., for nodes 615 and 630)in this node set, the mapping LUT would return two 42-bit strings, onefor Steiner tree 705 and one for Steiner tree 805. The bit string fortree 805 has values for paths 1 and 28 set to 1 and the remaining valuesset to 0, while the bit string for tree 705 will have values for paths 5and 26 set to 1 and the remaining values set to 0.

[0187] If the process did not retrieve more than one bit string at 2625,it transitions to 2635, which will be described below. Otherwise, whenthe process retrieves N bit strings (where N is an integer equal orgreater than 2) for N routes for routing the selected tree edge (i.e.,for connecting the selected pair of nodes), the process makes N−1duplicate copies of the current bit string or strings of the currenttree, and embeds the different routes among the copies of the tree.

[0188] In other words, the process makes (at 2630) N−1 duplicate copiesof the bit string or strings of the current tree; a bit string for thecurrent tree was initialized at 2610, and the current tree will havemultiple bit strings if its bit string was previously duplicated at2630. From 2630, the process transitions to 2635.

[0189] At 2635, the process modifies the bit string or strings for thecurrent tree with the bit string or strings (retrieved at 2625) for theselected tree edge. Next, the process determines (at 2640) whether ithas examined the last edge of the current tree (i.e., whether it hasexamined the last linked node pair in the current tree). If not, theprocess transitions back to 2620 to select the next tree edge (i.e., thenext linked node pair).

[0190] When the process determines (at 2640) that it has examined thelast tree edge, it then determines (2645) whether it has examined thelast tree supplied by the process 2200. If not, the process returns to2615 to select another tree and then determine the path-usage for thistree. Otherwise, the process transitions to 2650.

[0191] By the time the process 2600 reaches 2650, it has generatedbit-string representation of one or more trees. Each tree's bit-stringrepresentation is a 42-bit string. As mentioned above, onenode-representation of a tree might result in multiple bit-stringrepresentations when the tree's node set has one or more pairs of linkednodes that are not adjacent and one or more sets of paths exist betweenthe non-adjacent linked pairs.

[0192] In addition, a node configuration selected at 2215 might resultin different MST node representations (i.e., different node-representedMST's) that produce identical MST bit representations (ie., produceidentical bit-represented MST's). Accordingly, at 2650, the processexamines all the bit-string-represented trees and eliminates anyduplicate copy of the same bit-string-represented tree.

[0193] When a node configuration selected at 2215 results in a largenumber of bit-string-represented trees, the process 2600 can use abinary search tree (“BST”) to quickly sort and search the trees andthereby quickly identify and eliminate duplicate copies of the sametree. One such BST is described below by reference to FIGS. 32 and 33.

[0194] All the bit-represented trees that remain after 2650 are unique.Hence, after eliminating duplicate copies of the same trees, the process2600 stores (at 2655) all bit-string-represented trees that remain in astorage structure (such as a LUT). As described in the example below,each bit string specifies the route of a routing tree for the currentnode configuration. Specifically, as described in the example below,each stored bit string specifies the routing paths that a routing treefor the current node configuration traverses. At 2655, the processincrements each path value of the global count variable with eachbit-string-represented tree's corresponding path value, in order togenerate a total count value for each path. The process then recordsthis usage count for each path. Also, for each particular path, theprocess (at 2650) (1) divides the usage count by the number of the treesremaining after 2650 in order to obtain the usage probability value ofthe particular path, and then (2) stores this resulting probabilityvalue. The process then ends.

[0195] For the Steiner trees shown in FIGS. 6-8, the process 2600 wouldidentify three strings of 42-bits that specify the routing pathinformation for the three trees 605, 705, 805. These three bits stringsfor trees 605, 705, and 805 would respectively be:

000000000010000000000000000010000000110001;

000000000000000100000000010001000000100001;

000000000000010000000000010001000000000011

[0196] (In this document, the least significant bit (“LSB”) of abitstring is the rightmost bit, and the most significant bit (“MSB”) ofthe bitstring is the leftmost bit.) FIGS. 27 and 28 respectivelyillustrate examples of path-usage counts and path-usage probabilitiesfor the Steiner trees 605, 705, and 805 of FIGS. 6-8. In the discussionbelow, path-usage probability values are referred to as “probabilisticSteiner tree values.”

[0197] 3. Retrieving Steiner Trees

[0198] When the Steiner-tree routes are pre-tabulated according toprocess 2200, a router at run-time identifies one or more Steiner-treeattributes (e.g., routes) for a net in the following manner. The routerfirst identifies the net's configuration with respect to thepartitioning grid. It then uses the identified configuration to retrieveone or more attributes (e.g., routes) that are stored for the identifiedconfiguration in the storage structure.

[0199] In some embodiments, the storage structure is a look-up table(“LUT”) of floating point numbers. In some of these embodiments, the LUTis indexed by a configuration code. In other words, to retrieve aparticular attribute for a particular net configuration, theconfiguration code for the net configuration is identified, and thisconfiguration code is used to identify the entry in the LUT that storesthe desired attribute.

[0200] Some embodiments use the octagonal wiring model illustrated inFIG. 3, and specify each net's routing path in terms of the 42 diagonaland Manhattan routing paths illustrated in FIG. 12. In some of theseembodiments, the LUT stores 42-bits for each route, where each bitrepresents one of the 42 paths. Also, each net's configuration code is a16-bit number, where each bit represents a sub-region defined by the 4×4partitioning grid. Each configuration-code bit is set (e.g., equals 1)when the associated net has a pin in the sub-region represented by theconfiguration-code bit, and is not set (e.g., equals 0) when theassociated net does not have a pin in this sub-region. Also, in theseembodiments, there are 2¹⁶ configuration codes that represent the 2¹⁶possible net configurations.

[0201] For instance, the net configuration code is 000001000000001, whenthe net has a pin in slots 0 and 9. For such a configuration, someembodiments pre-tabulate two trees, one that uses paths P17 and P24, andanother that uses paths P12 and P30. Each of these trees can bespecified by a string of 42 bits. The bit string for the first tree is,

000000000000000001000000100000000000000000,

[0202] while the bit string for the second tree is,

000000000001000000000000000001000000000000.

[0203] Some embodiments store these two bit strings in a LUT, andretrieve these two bit strings by using the 16-bit configuration code ofthe net configuration 0000001000000001.

[0204] 4. Storing Steiner Trees in a Compressed Form

[0205] A variety of compression techniques can be employed to store anduse Steiner-tree routes for sets of net configurations. One suchtechnique is illustrated in FIG. 29. The process 2900 of this figure issimilar to the process 2200 described above, except that the process2900 has two additional operations 2905 and 2910, and has slightlydifferent operations 2215 and 2250. Operations 2205, 2210, and 2220-2245of process 2900 are identical to similarly numbered operations 2205,2210, and 2220-2245 of process 2200. Accordingly, these operations 2205,2210, and 2220-2245 will not be further described below, in order not toobscure the description of the invention with unnecessary detail.

[0206] The process 2900 performs the additional operations 2905 and 2910to reduce the amount of information that is pre-tabulated. The firstoperation 2905 reduces the number of potential net configurations forwhich the process 2900 pre-tabulates routes, while the second operation2910 ensures that the process 2900 stores each Steiner-tree route onlyonce.

[0207] Both of these operations are further described below. One ofordinary skill, however, will realize that some embodiments do not useboth these operations. For instance, some embodiments might only perform2910 to ensure that each Steiner-tree route is only stored once.

[0208] a. Symmetrical Net Configurations

[0209] The operation 2905 groups the potential net configurationidentified at 2210 into sets of symmetrical net configurations. From2215-2250, the process 2900 then generates and stores one set of Steinertrees for one designated net configuration of each group of symmetricalconfigurations. This flow is directed by 2215 and 2250. At 2215, theprocess 2900 selects a designated node configuration that it has notpreviously examined. At 2250, the process determines whether it hasexamined the designated node configuration for each group of symmetricnode configurations. At run-time, the designated configuration of eachgroup directly uses the pre-tabulated routes for its group, while thenon-designated configurations of each group generate their routes fromthe pre-tabulated routes for their group.

[0210]FIGS. 30 and 31 illustrate one technique for performing thisgrouping. This technique is performed for FIG. 5's 4×4 partitioninggrid. In this grid, each net configuration is symmetrical with respectto seven other net configurations. These seven symmetricalconfigurations can be identified by (1) rotating the net configurationby90°, (2) rotating it by 180°, (3) rotating it by 270°, (4) flippingthe net configuration about the x-axis, (5) rotating the netconfiguration by 90° and flipping the result about the x-axis, (6)rotating it by 180° and flipping the result about the x-axis, (7)rotating it by 270° and flipping the result about the x-axis.

[0211] In the embodiments described below, the rotation and flipoperations are defined with respect to a Cartesian coordinate systemthat has (1) an x-axis parallel to the width of the 4×4 partitioninggrid (i.e., width of the layout), (2) a y-axis parallel to the height ofthe grid, and (3) an origin at the intersection of grid's slots 5, 6, 9,and 10, which are illustrated in FIG. 5. Specifically, the rotation isdefined in terms of a clockwise rotation about the origin. The flippingof a configuration involves changing the sign of the y-coordinates ofeach configuration slot. Table 1 below illustrates an example of eightnet configurations that are related based on the above-describedsymmetrical relationship. TABLE 1 Configuration Slots With PinsDescription of Symmetry 0000000100000001 0, 9 Original configuration0000101000000000 10, 12 Rotate by 90° 1000000000100000  6, 15 Rotate by180° 0000000000101000 3, 5 Rotate by 270° 0000001000001000  5, 12 Flipabout x-axis 1000000100000000 0, 6 Rotate by 90° and flip about x-axis0001000000100000  3, 10 Rotate by 180° and flip about x-axis0000000001000001  9, 15 Rotate by 270° and flip about x-axis

[0212]FIG. 3 illustrates a process 3100 for grouping net configurationsaccording to the symmetries described above. FIG. 30 illustrates fourdata fields that the process 3100 stores for each configuration. Thefirst field 3000 stores the configuration's 16-bit pin distribution(i.e., its net/node configuration). The second field 3005 specifieswhether the process 3100 has already grouped the configuration withother configurations.

[0213] The third field 3010 is a reference (e.g., a pointer) to atreelist 3020, which includes one or more references to one or moreSteiner-tree routes 3025 for the configuration's group. Eachconfiguration in a group refers to the same treelist 3020. For instance,FIG. 30 illustrates three grouped configurations 3030, 3035, and 3040that refer to the same treelist. The fourth field 3015 stores thesymmetrical-relation identifier. This identifier specifies how to obtaintrees for the net configuration from the trees stored for the group. Inother words, each configuration's identifier specifies how to transformone or more trees that are pre-tabulated for the configuration's groupinto one or more trees for the configuration.

[0214] The process 2900 performs process 3100 at 2905 after the process2900 defines (at 2210) all sets of potential node configurations. Asshown in FIG. 31, the process 3100 initially selects (at 3105) one ofthe node configurations that was defined at 2210. It then marks (at3110) this configuration as grouped in its configurations field 3005.

[0215] Next, the process records (at 3115) “NONE” in thisconfiguration's relation-identifier field 3015. This marking indicatesthat the pre-tabulated trees specified for this configuration (i.e., thetrees that will be referred to by this configuration's treelist 3020) donot need to be transformed in any manner for the selected nodeconfiguration. In each group of configurations, the configuration thathas “NONE” recorded in its relation-identifier field is the designatedconfiguration for the group (i.e., it is the configuration that candirectly use the Steiner trees that are generated for the group).

[0216] At 3120, the process then creates a treelist 3020 for thisconfiguration's group, and links this configuration's reference field3010 to this treelist. To this treelist, the process 2900 will add (at2910) references that refer to the trees for this configuration's group.

[0217] The process 3100 then selects (at 3125) one of the sevensymmetrical relationships described above. It next uses (at 3130) theselected symmetrical relationship to identify one of the sevenconfigurations that are symmetrically related to the configurationselected at 3105. Some embodiments have seven LUT's, one for eachsymmetrical-transform relationship. Each LUT provides a one-to-onemapping that specifies a symmetrical node for each potential node of adesignated node configuration. For instance, Table 2 below identifiesthe corresponding nodes for the symmetrical configuration that can beobtained by rotating the designated node configuration by 90°. TABLE 2Node of Designated Corresponding Node of the 90°- Configuration RotatedSymmetrical Configuration Slot 0  Slot 12 Slot 1 Slot 8 Slot 2 Slot 4Slot 3 Slot 0 Slot 4  Slot 13 Slot 5 Slot 9 Slot 6 Slot 5 Slot 7 Slot 1Slot 8  Slot 14 Slot 9  Slot 10  Slot 10 Slot 6  Slot 11 Slot 2  Slot 12 Slot 15  Slot 13  Slot 11  Slot 14 Slot 7  Slot 15 Slot 3

[0218] At 3135, the process then marks the configuration identified at3130 as grouped in the configuration's field 3005. It next records (at3140) the identity of the relationship selected at 3125 (e.g., rotatedby 90°) in the configuration's relationship-identifier field 3015. Thisoperation can be used at run-time to transform one or more trees thatare pre-tabulated for the configuration's group into one or more treesfor the configuration identified at 3130.

[0219] The process then links (at 3145) the reference field 3010 of theidentified configuration to the treelist 3020 for this configuration'sgroup. At 3150, the process then determines whether it has generated allseven configurations that are symmetrically related to the one selectedat 3105. If not, the process selects (at 3125) another symmetricalrelationship, and then performs 3130-3145 to identify the relatedconfiguration and populate its group fields.

[0220] When the process determines (at 3150) that it has generated allseven configurations related to the configuration selected at 3105, itdetermines (at 3155) whether it has examined all node configurationsthat the process 2900 generated at 2210 (i.e., whether it has marked allgenerated node configurations as “grouped”). If not, the processtransitions to 3105 to select a node configuration that has not yet beenmarked as “grouped,” and repeats the above-described operations for thenewly selected configuration and its symmetrically-relatedconfigurations. When the process determines (at 3155) that it hasexamined all node configurations, it ends.

[0221] b. Storing Each Tree Only Once

[0222] At 2910, the process ensures that the process 2900 stores eachSteiner-tree route only once for any node configuration that might usesuch a route. Like process 2200 of FIG. 22, the process 2900 calls (at2245) the process 2600 to calculate the routing-path information for theSteiner tree or trees that the process 2900 identified for a nodeconfiguration. The process 2600 identifies one or more bit strings torepresent each Steiner tree identified by process 2900. The process 2600also (1) eliminates (at 2650) any duplicate copies of eachbit-represented tree that it generates for the same node configuration,and then (2) stores (at 26555) each remaining bit-represented tree.

[0223] However, when the process 2600 works in conjunction with process2900, it does not permanently store (at 2655) each generated bit string.Instead, it returns the generated bit strings to process 2900. Theprocess 2900 then checks (at 2910) whether it previously stored eachreturned bit string (i.e., each returned Steiner tree) in the storagestructure for previous node configuration (i.e., for a nodeconfiguration that was previously selected at 2215). If so, the processdoes not re-store this bit string, but rather links one of thereferences in the node configuration's treelist 3020 to thepreviously-stored bit string. If not, the process stores this bit stringin the storage structure 3050 of FIG. 30 and links one of the referencesin the node configuration's treelist 3020 to the newly-stored bitstring.

[0224] A variety of different techniques can be used to check (at 2910)whether the process 2900 previously stored a bit string in the storagestructure 3050. The embodiments described below use a binary-search treeto perform this checking operation.

[0225]FIG. 32 illustrates one such binary-search tree (“BST”). This tree3200 has numerous nodes 3220, with each node having zero or two childnodes. Each node in the tree includes two references 3205 and 3210 forreferring to the node's left and right child nodes. Each node also has areference 3215 for referring to a 42-bit Steiner tree that correspondsto the node.

[0226] The BST has forty-two levels, where each level corresponds to oneof the bits in the 42-bit string for representing Steiner trees. The BSTlevels are in the same order as the bits in the bit string. Accordingly,the BST's 0^(th) level corresponds to the string's 0^(th) bit (i.e., thebit corresponding to path 0), the BST's 1^(st) level corresponds to thestring's 1^(st) bit (ie., the bit corresponding to path 1), the BST's2^(nd) level corresponds to the string's 2^(nd) bit (i.e., the bitcorresponding to path 2), etc. At each level, the value of the stringbit corresponding to that level determines the branching.

[0227]FIG. 33 illustrates a process 3300 that the process 2900 uses (at2910) to traverse the BST 3200 to determine whether a Steiner tree waspreviously stored in the storage structure. As shown in FIG. 33, theprocess 3300 initially sets (at 3305) a variable L to 0. This variablespecifies the BST's level that the process 3300 is currently examining.At 3310, the process determines whether the L^(th) bit in the bit stringis a 0. If not, the process (at 3315) increments the variable L by one,and defines the current node's left child node as the current node. Ifso, the process increments (at 3320) the variable L by one, and definesthe current node's right child node as the current node.

[0228] From 3315 or 3320, the process transitions to 3325. Here, theprocess determines whether it has examined all the bits in the bitstring, and if not, whether all the remaining unexamined bits (i.e., theL^(th) bit to the 41^(st) bit of the bitstring) are 0. If all the bitshave not been examined and one or more of the unexamined bits have avalue of 1, the process returns to 3310 to examine the current node.

[0229] On the other hand, if all the bits have been examined or all ofthe unexamined bits are 0, the process has found the node that shouldstore the bit string. Accordingly, the process determines (at 3330)whether the current node's tree reference 3215 refers to a stored tree(i.e., a stored bit string). If not, the process stores (at 3335) thebit string in the storage structure 3050 and links the current node'stree reference 3215 to this structure. The process also links (at 3335)one of the references in the node configuration's treelist 3020 to thenewly-stored bit string. If the process determines (at 3330) that thecurrent node's tree reference 3215 refers to a previously-stored bitstring, the process just links (at 3340) one of the references in thenode configuration's treelist 3020 to the previously-stored bit string.After 3335 or 3340, the process ends.

[0230] c. Identifying Routes From the Compressed Pre-Tabulated Table

[0231] When the Steiner-tree routes are pre-tabulated according toprocess 2900, a router at run-time identifies one or more Steiner-treeroutes for a net in the following manner. The router first identifiesthe net's configuration with respect to the partitioning grid. From thestorage structure 3050, it then retrieves one or more routes 3025specified by the treelist 3020 of the identified configuration.

[0232] The process then identifies the symmetrical relationship betweenthe identified net configuration and the designated configuration forits group. It next uses this relationship to identify one or more routesfor the identified net configuration from the retrieved routes. To dothis, some embodiments use seven LUT's, one for eachsymmetrical-transform relationship. Each LUT provides a one-to-onemapping that specifies a path that is symmetrical to each potential paththat a route of the designated node configuration can use.

[0233] For instance, the net configuration might be 0001010000000000,which indicates the net having a pin in slot 10 and 12. Thisconfiguration is symmetrically related to the net configuration0000001000000001, which indicates the net having a pin in slot 0 and 9.Specifically, the configuration 0001010000000000 is obtained when theconfiguration 0000001000000001 is rotated by 90°.

[0234] When the net configuration 0000001000000001 is the designatedconfiguration, some embodiments pre-tabulate two trees, one that usespaths P17 and P24 and another that uses paths P12 and P30. By rotatingthese tees by 90°, the router can identify two routes for theconfiguration 0001010000000000. To rotate each tree by 90°, the router(1) identifies each path used by the tree (i.e., identifies each set bitin the 42-bit string specifying the tree), and (2) from the 90°-rotationLUT, identifies the paths that are symmetrically related to theidentified paths for a 90° rotation of the partitioning grid.

[0235] Accordingly, from the 90°-rotation LUT, the router identifiespath 37 as the path related to path 24 through a 90° rotation, andidentifies path 7 as the path related to path 17 through a90° rotation.From the 90°-rotation LUT, the router identifies path 9 as the pathrelated to path 12 through a 90° rotation, and identifies path 39 as thepath related to path 30 through a90° rotation. In this manner, therouter identifies two trees for the configuration 0001010000000000. Onetree uses paths 7 and 37, and the other uses paths 9 and 39.

[0236] One of ordinary skill in the art will realize that otherembodiments do not identify trees for symmetrical-node configurations byusing LUT's. For instance, some embodiments might mathematicallyidentify trees for symmetrical-node configurations. For each symmetricalrelationship, these embodiments might use a different mathematicalequation to map the paths of the pre-tabulated tree to the paths of thesymmetrically-related tree.

[0237] 5. Pre-Tabulating Steiner Trees for Different Wiring Models.

[0238] Some embodiments of the invention pre-tabulate several sets ofSteiner trees for several different wiring models. For instance, FIG. 34illustrates a process 3400 that performs the process 2200 or process2900 (1) once (at 3405) for a wiring model that has horizontal,vertical, and ±45° lines, (2) once (at 3410) for a wiring model that hashorizontal, vertical, and ±120° lines, and (3) once (at 3415) for awiring model that has horizontal and vertical lines.

[0239] To model all possible net configurations for a wiring model thatuses horizontal, vertical, and ±45° lines, this process calculates (at3405) the length, routing path, and path-usage values of Steiner treeswith potential 45° diagonal edges. In other words, at 3405, the process3400 uses 45° as the angle A in Equation (A) that is used by processes2400 and 2500 of process 2200 or 2900.

[0240] To model all possible net configurations for a wiring model thatuses horizontal, vertical, and ±120° lines, this process calculates (at3410) the length, routing path, and path-usage values of Steiner treeswith potential 120° diagonal edges. In other words, at 341 0, theprocess 3400 uses 120° as the angle A in Equation (A) that is used byprocesses 2400 and 2500 of process 2200 or 2900.

[0241] To model all possible net configurations for a wiring model thatuses horizontal and vertical lines, these embodiments calculate (at3415) the length, routing path, and path-usage values of ManhattanSteiner trees. In other words, at 3415, the process 3400 uses 90° as theangle A in Equation (A) that is used by processes 2400 and 2500 ofprocess 2200 or 2900.

[0242] B. Pre-Tabulating and Generating Trees

[0243] Some embodiments use net configurations to retrieve and generateroutes. For instance, some embodiments use net configurations toretrieve pre-tabulated routes for certain nets while generating routesfor other nets. Several such embodiments will now be described byreference to FIGS. 35-38.

[0244] These embodiments pre-tabulate routes, referred to below as“minimum closed trees” or MCT's, for closed node configurations. An MCTis a MST for a close node configuration. In other words, an MCT is aminimal tree that has N−1 edges that span the N nodes of theconfiguration through the shortest route, which only branches (i.e.,starts or ends) at the nodes. For open node configurations, theseembodiments pre-tabulate certain related closed node configurations. Theterms closed node configurations and open node configurations will bedescribed below by reference to FIGS. 35A and 35B.

[0245]FIG. 35A illustrates an example of a closed node configuration3505 (including nodes 3515, 3520, 3530, 3535, and 3540), while FIG. 35Billustrates an example of an open node configuration 3510 (includingnodes 3515, 3530, 3535, and 3540). The node configuration 3505 is aclosed one since all the nodes in this configuration are adjacent to atleast one other node in the configuration. The node configuration 3510is an open one since node 3515 is not adjacent to another node in theconfiguration; in this configuration, node 3515 has node 3545 between itand the next closest node 3530.

[0246] Node configuration 3510 has several related closed nodeconfigurations. Two such configurations that do not result in MCT's withan “antenna” node are (1) a first configuration that includes 3515,3545, 3530, 3535, and 3540, and (2) a second configuration that includes3515, 3550, 3530, 3535, and 3540. The first configuration is obtained byadding node 3545 to configuration 3510, while the second configurationis obtained by adding node 3550 to configuration 3510.

[0247] A configuration that is obtained by adding node 3555 and eithernode 3545 or 3550 to the configuration 3510 is a related closedconfiguration that will always result in MCT's that have node 3555 as anantenna node. An antenna node in an MCT of a closed node configurationthat is obtained by adding one set of nodes to an open nodeconfiguration, is a node that is part of the added set and that has onlyone of the MCT's edges incident upon it. As further described below, thefirst two node configurations (configuration 3515, 3545, 3530, 3535, and3540, and configuration 3515, 3550, 3530, 3535, and 3540) are relatedclosed node configurations that can be pre-tabulated for the open nodeconfiguration 3515, 3530, 3535, and 3540. The third configuration (3515,3530, 3535, 3540, and 3555), on the other hand, should not bepre-tabulated for this open configuration since it leads to an antennanode.

[0248] 1. Pre-tabulating MCT's

[0249]FIG. 36 illustrates a process 3600 that pre-tabulates MCT's forall node configurations within a particular partitioning grid, such asthe 4-by-4 grid of FIG. 5. As shown in FIG. 36, the process 3600initially identifies (3605) all potential node configurations.

[0250] It then selects (at 3610) a configuration. If the selectedconfiguration is a closed one, the process next identifies (at 3615) allMCT's for the selected configuration. As mentioned above, a nodeconfiguration is a closed one if each node in the configuration isadjacent to at least one other node in the configuration. One ofordinary skill will appreciate that the process 3600 can identify eachMCT for the selected configuration directly based on the sets of paths(e.g., the 42 paths illustrated in FIG. 12) that exist between the nodesof the selected closed node configuration. Each MCT is a uniquecombination of N−1 of such paths that connect all N nodes of the closednode configuration through the shortest route. Such MCT's can beidentified through a recursive operation that explores all shortestpaths between nodes of the closed configuration; in some embodiments,such a recursive operation would be similar to the one explained aboveby reference to FIGS. 25A and 25B, except that it specifies each MCTdirectly based on the interconnecting paths.

[0251] At 3615, the process also computes the cost of each MCTidentifies at 3615. In some embodiments, the process computes each MCT'scost by (1) assigning costs to the Manhattan and diagonal paths thatconnects the adjacent slots of the partitioning grid, (2) identifyingthe paths used by the MCT, and (3) summing up the path costs. A nodeconfiguration that has less than two nodes does not have a MCT, andaccordingly its MCT cost is zero.

[0252] After 3615, the process determines (at 3620) whether it hasexamined all the configurations. If not, the process returns to 3610 toselect another one. Otherwise, the process ends.

[0253] 2. Pre-tabulating Related Closed Node Configurations

[0254]FIG. 37 illustrates a process 3700 that, for an open nodeconfiguration, pre-tabulates certain related closed node configurations.This process initially identifies (at 3705) candidate sets of connectionnodes. Each candidate set does not include any of the nodes of the opennode configuration. Also, the candidate sets include all possible nodeconfigurations that can be obtained without the nodes of the open nodeconfiguration.

[0255] Next, the process selects (at 3710) a candidate set of connectionnodes. The process then determines (at 3715) whether the combinedconfiguration, resulting from the addition of the selected candidate setand the open node configuration, has one or more pre-tabulated MCT's. Ifnot, the process transitions to 3745, which is described below.

[0256] If the combined configuration has one or more pre-tabulatedMCT's, the process then performs 3720-3740 to determine whether thecombined node configuration results in at least one MCT without antennanodes. If all the MCT's have an antenna nodes, then the combinedconfiguration obtained for the candidate connection node selected at3710 is not stored as a related closed configuration for the open nodeconfiguration.

[0257] Specifically, at 3720, the process selects one of the MCT's ofthe combined configuration. It then identifies (at 3725) all nodes ofthis MCT that have a degree 1 (i.e., all nodes that have only one of theMCT's path incident upon them). Next, the process determines (at 3730)whether all the identified nodes are part of the open nodeconfiguration. If so, the process accepts the combined configurationobtained with the candidate set selected at 3710, and stores (at 3740)the combined configuration (obtained by combining the candidate setselected at 3710 with the open node configuration) as a related closednode configuration for the open node configuration. From 3740, theprocess transitions to 3745, which will be described below.

[0258] On the other hand, if the process determines (at 3730) that atleast one of the node identified at 3725 is not part of the open nodeconfiguration, the process determines whether it has examined all theMCT's for the combined node configuration. If not, the process returnsto 3720 to select another MCT. Otherwise, the process transitions to3745. At 3745, the process determines whether it has examined all thecandidate set of connection nodes. If not, the process transitions backto 3710 to select and examine another candidate set. Otherwise, theprocess ends.

[0259] 3. Generating MCT's During Run-Time

[0260] When the routes and closed node configurations are pre-tabulatedaccording to processes 3600 and 3700, a router at run-time identifiesone or more routes for a net according to the process 3800 of FIG. 38.As shown in this figure, the process first identifies (at 3805) thenet's configuration with respect to the partitioning grid.

[0261] The router then determines (at 3810) whether the storagestructure stores one or more MCT's for the identified configuration. Ifso, the process retrieves (at 3815) the stored MCT's for theconfiguration and stores them in the list of routes for the net. After3815, the process then ends.

[0262] On the other hand, if the process determines (at 3810) that thestorage structure does not store any MCT's for the identifiedconfiguration, the process retrieves (at 3820) the related closed nodeconfigurations for the identified configuration. It then selects (at3825) one of the retrieved closed configurations.

[0263] Next, the process retrieves (at 3830) the MCT's for the selectedclosed configuration. It then determines (at 3835) whether to store theretrieved MCT's for the identified node configuration. In someembodiments, the process makes this determination based purely on thewirelength cost of the MCT's. In other words, in these embodiments, theprocess stores the MCT's only if they are the shortest MCT's that theprocess has examined thus far for the identified node configuration. Inother embodiments, however, the process decides whether to store theMCT's based on other factors such as the estimated congestion of therouting paths, the estimated number of vias, the number of MCT'sselected thus far, etc. Also, in some embodiments, the process 3700sorts an open node configuration's closed node configurations in aparticular order. For instance, the process 3700 might sort the closedconfigurations that produce the shorter MCT's first. In theseembodiments, the process 3800 examines the closed node configurationsaccording to the stored order, and once it identifies the R number ofroutes, the process 3800 terminates.

[0264] If the process decides not to store the retrieved MCT's, theprocess transitions to 3845, which is described below. However, if theprocess decides (at 3830) to store the retrieved MCT's, its stores theseMCT's in a list of routes for the net configuration. The process thendetermines (at 3840) whether it has examined all the related closed nodeconfigurations for the identified net configuration. If not, the processreturns to 3820 to select another closed configuration; in someembodiments, the process might not return to 3820 to select anotherclosed configuration if it has already identifies a certain number ofMCT's. When the process determines that it has examined all the relatedclosed node configurations, the process ends.

VI. Recursive 4-by-4 Partitioning Router

[0265] A. Software Architecture.

[0266]FIG. 39 illustrates the software architecture of a router 3900 ofsome embodiments of the invention. This router can operate with avariety of different wiring architectures. It can also operate withdifferent partitioning, congestion, and path-defining grids. However, inthe embodiments described below, the router is primarily described towork in conjunction with (1) the octagonal wiring model that isillustrated in FIG. 3, (2) the partitioning grid that is illustrated inFIG. 5, and (3) the congestion grids and their associated 42 paths thatare illustrated in FIGS. 9-12.

[0267] The software architecture of FIG. 39 includes several softwaremodules 3905 and several data constructs 3910. The software modulesinclude an initializer 3915, a slot manager 3925, a solver 3930, apropagator 3935, a saver 3940, a linear-programming (“LP”) solver 3945,an integer-linear-programming (“ILP”) converter 3950, while the dataconstructs 3910 include LUT's 3965, circuit modules 3970, net list 3972,nets 3974, slots 3976, slot-nets 3978, paths 3980, and pins 3982.

[0268] The router 3900 defines partitioning grids that recursivelydivide a design region (i.e., the IC layout or a region of the IClayout)_into smaller and smaller sub-regions. In the embodimentsdescribed below, the router uses 3 evenly-spaced horizontal lines and 3evenly-spaced vertical lines to recursively divide the design regionsinto 16 identically-sized sub-regions (i.e., 16 identically-sizedslots). FIG. 40 illustrates a design region 4005 that is recursivelydivided into sets of 16 sub-regions. Specifically, the design region isdivided initially into 16 sub-regions, each of these sub-regions isfurther divided into 16 smaller sub-regions, and one of the smallersub-regions 4010 is further sub-divided into 16 sub-regions. At eachrecursion level, the router simply adjusts the coordinates of thepartitioning grid to match the coordinates of the IC region at thatrecursion level. In other embodiments, the router can use differentshaped partitioning grids for all or some recursion levels.

[0269] The router 3900 defines in a hierarchical, top-down manner thewiring path values for each net in the design region. The router'sinitializer 3915 initially determines the number of recursion levels,and the number of slots resulting from this number of recursion levels.The initializer also creates data structures for these slots. Inaddition, for each slot, the initializer creates a slot-net datastructure for each net in the slot, and this slot-net data structurestores the net's configuration within that slot. For each slot, theinitializer also identifies all the circuit modules that intersect thisslot.

[0270] In some embodiments, the initializer also defines the capacitiesof routing paths within each slot and stores these capacities in theslot's data structure. In the embodiments described below, however, theslot manager defines these capacities as it directs the routing of eachslot.

[0271] For each slot that is partitioned into smaller child slots, theslot manager directs the solver 3930 to select a route for each net thathas actual or virtual pins in the slot. The solver uses each net'sconfiguration (1) to identify one or more optimal routes for each net,and at times (2) to generate fake configuration to identify one or moresub-optimal routes for each net. The solver identifies one or moreroutes for a particular configuration based on any one of the approachesdescribed above in Section V.

[0272] The solver then formulates an LP problem and feeds thesesolutions to the LP solver 3945, which, in turn, returns a number ofreal-number solutions. These real-number solutions are then convertedinto integer solutions by the ILP solver 3950. These integer solutionsspecify a particular route for each net, and the solver stores eachnet's route information in the net's slot-net data structure for thecurrent slot.

[0273] After the solver specifies the route for each net that has anactual or virtual pin in the current slot, the slot manager 3925 callsthe propagator 3935 if the current slot is not a leaf slot. A leaf slotis a slot that has child slots, but its child slots do not have anychild slots (i.e., its child slots are not partitioned). The child slotsof a leaf-level slot are called Gcells.

[0274] When called by the slot manager, the propagator determines howthe routing paths specified by the solver for the current routing levelpropagate down into the child slots of the current slot. For slots thatare after the top-level slot and before the leaf-level slot, thepropagator also performs a follow-up propagation operation thatpropagates the routing paths specified by the propagator at the previousrouting level one level further down.

[0275] For each net in the current slot, the propagator has to determinethe net's pin distribution within all child slots affected by the net'srouting paths. The propagation process often entails adding virtual pinsin the current slot's grandchild slots (i.e., the child slots of thecurrent slot's child slots). In other words, the propagator might modifythe net's configuration within the child slots of the current slot.

[0276] Different embodiments of the invention use different propagators.Two different propagators are described below. The first propagatorenumerates several propagation solutions for each net's route and thenuses the LP solver 3945 and ILP converter 3950 to select a propagationsolution for each net. The second propagator, on the other hand, is asequential propagator that uses a greedy approach to select and embed apropagation for the route of each net in the current slot. In theembodiments described below, both these propagators use a sequentialpropagator to perform the follow-up propagation, when applicable.

[0277] The identified propagations for each net's route specify aparticular configuration for the net within each affected child slot,and the propagator stores the net's configuration in the net's slot-netdata structure for the affected child slots. The embodiments describedbelow specify each net's configuration by a string of 16 bits, whereeach bit corresponds to a child slot of a slot.

[0278] After the solver has specified the route for each net in a leafslot, the slot manager calls the saver 3940 to link the path structuresof each net's route to their respective net's main data structures. Thesaver also links to the main net data structures the path datastructures of the propagation paths that the propagator specifies for aparent slot of a leaf-level slot (i.e., for a grandparent slot of aGcell). These propagation paths include paths that the propagatoridentified (1) for the routing paths specified by the solver and (2) forthe routing paths specified by the propagator at the previous routinglevel.

[0279] In this manner, the path data structures linked to a net's maindata structure collectively represent the final route for the net thatthe router specifies. In some embodiments, such a route is a globalroute for a net. FIGS. 49-83 further describe the software modules 3905.However, before describing these software modules, the data constructs3910 will be described below by reference to FIGS. 41-48.

[0280] B. Data Constructs.

[0281] 1. LUT's.

[0282] The LUT's 3965 store information (such as routing paths, lengths,path-usage, etc.) about the routes that connect the sub-regionscontaining the circuit modules of the nets. Some of these routes haveedges that are completely or partially diagonal. In the embodimentsdescribed below, the LUT's store the length, routing paths, andpath-usage probability values of the routes for all possible netconfigurations. Several processes for selecting routes andpre-tabulating their length, routing paths, and path-usage probabilitieswere discussed above in Section V. One of ordinary skill will understandthat other embodiments also store other attributes of trees. In theembodiments that utilize the route-generating process 3800 of FIG. 38,the LUT's store for each open node configuration one or more relatedclosed node configurations.

[0283] In some embodiments, the router 3900 can operate with differentwiring architectures. In these embodiments, different LUT's can be usedto store route attributes for the different wiring models. For instance,the LUT's can store routing information for (1) a first wiring modelthat uses Manhattan and ±45° diagonal lines, (2) a second wiring modelthat uses Manhattan and ±120° diagonal lines, (3) a third wiring modelthat only uses Manhattan lines, etc. Once the wiring model is selected,the routing information for each net configuration can be retrieved fromthe LUT that is appropriate for the selected wiring model.

[0284] In the embodiments where the router can operate with differentwiring models, the router typically selects the wiring model at thebeginning of the design process. For instances, in some embodiments, theprocess 400 selects the wiring model at 405 before it selects thepartitioning grid. Also, some embodiments might switch from one wiringmodel to another for different portions of the design process or atlower levels of the design hierarchy.

[0285] In the embodiments described below, the router uses the octagonalwiring model that is illustrated in FIG. 3 throughout the routingprocess. The router uses a LUT that stores routing information (e.g.,routes, lengths, path-usage values, trees for closed-sets of slots, setsof nodes, etc.) for the congestion grids and their associated 42 pathsthat are illustrated in FIGS. 9-12.

[0286] 2. Net List, dbNet, Slot-Net, Pin, and Path Structures.

[0287]FIG. 41 illustrates the data structure for a net list 4100. Thislist includes one or more fields 4105, each of which refers (e.g.,points) to a dbNet data structure 4110. Each net has a dbNet datastructure, which serves as the main data structure for the net.

[0288]FIG. 42 illustrates the dbNet data structure. This data structureincludes an index field 4205 that contains a value that some of thesoftware modules (e.g., the propagator) use to identify the net. Thisdata structure also includes a number of fields 4210 that refer (e.g.,point) to pin data structures.

[0289]FIG. 43 illustrates a simple pin data structure 4300 that includesa location field that specifies the pins location. In some embodiments,the pin's location is provided as a three-dimensional location that notonly specifies its x and y location, but also specifies its layer. Otherembodiments, however, store the pin's layer as part of a pin macro. Thispin macro can be stored as part of the circuit macro which is referencedby the slot-data structure as described below.

[0290] The dbNet data structure also includes one or more fields 4220,each of which refers (e.g., points) to a path data structure 4400 suchas the one illustrated in FIG. 44. In some embodiments, the saver linksthe path data structures that specify the final lowest-level route pathsfor each net to the dbNet of their respective nets through referencefield 4220.

[0291] The path data structure includes a field 4405 that specifies thepath type as horizontal (H), vertical (V), a first diagonal direction(E), or a second diagonal direction (W). This structure also includes afield 4410 to store the path id, which is a number from 0 to 41 thatidentifies the data structure's path as one of the 42 paths defined inthe two-grids. In addition, this data structure includes a field 4415that refers (e.g., points) to the dbNet associated with the path.Finally, this data structure has two fields 4420 that refer to the datastructures of the two slots that the path is incident upon. These twofields can be used during a verification process to ensure thecontinuity of the routes specified by the router 3900.

[0292] For each slot, the router 3900 defines a slot-net data structurefor each net that has an actual or virtual pin in the slot, and thisslot-net data structure stores the net's configuration within that slot.FIG. 45 illustrates this slot-net data structure. This structureincludes a field 4505 that refers (e.g., points) to the dbNet of itsnet. It also includes a field 4510 that stores a bit string thatspecifies its net's pin distribution in the slot. As discussed furtherbelow, the initializer initially sets this field based on all the actualpins of the net within the slot. During the recursion process, thepropagator might modify the bit string in this field 4510 to account forvirtual pins. The slot-net data structure also includes a field 4515that stores the 42-bit selected-route string. The solver sets this bitstring after it selects a route for net in the slot.

[0293] 3. Slot.

[0294] The router 3900 recursively divides the design region into setsof 16 sub-regions or slots. FIG. 46 presents a graph that conceptuallyillustrates the hierarchy of slots (i.e., sub-regions) defined by therouter. This graph 4600 illustrates two levels 4610 and 4615 of therecursion process. In this graph, each node represents an IC region at aparticular stage within the recursion process. Also, in this graph, theroot node represents the entire design region, while each non-root noderepresents a portion of the design region.

[0295] In a slot-hierarchy, each node has either 0 child nodes or 16child nodes. A node has 16 child nodes when the router partitions thatnode's region into 16 sub-regions. Conversely, a node does not havechild nodes when its corresponding region is not partitioned.

[0296] In some embodiments, the router 3900 defines a slot datastructure to represent each node in a slot-hierarchy. FIG. 47 presentsone such data structure 4700 for a slot. This data structure specifiesthe coordinates 4710 of the slot. It also includes a reference (e.g., apointer) 4720 to a list 4740 of circuit modules in the slot. This list4740 includes one or more references (e.g., one or more pointers) 4745to one or more circuit modules 4800 in the slot.

[0297] The slot data structure 4700 also includes a reference to a list4730 of slot-nets of the slot. The list includes references 4735 toslot-net data structures 4500. In the embodiments described below, theslot data structure does not have references to its child slots. Some ofthese embodiments order the slots in a pre-specified order on a list,and based on this order, these embodiments identify corresponding childand parent slots. The slot data structure 4700 also includes a field4725 that specifies the slot's unique identifier.

[0298] 4. Circuit Modules.

[0299]FIG. 48 illustrates the data structure 4800 of a circuit module.This data structure stores the orientation (4805) and the position(4810) of the circuit module. It also includes a reference 4815 to thecircuit macro 4820, which contains a description of the circuit module.For instance, the circuit macro refers to data structures 4825 forobstacles within the circuit module. The obstacle data structuresspecify information about the obstacles, such as the layer (4830) andshape (4835) of the obstacles.

[0300] C. Initializer.

[0301]FIG. 49 illustrates a process 4900 performed by the initializer atthe start of the routing operation. Before this process starts, therouter typically has received a placed net list, technology definition(including number of layers, preferred wiring direction for each layer,routing pitch for each layer, etc.), and the number of tracks for thelowest level slots (i.e., for the Gcells).

[0302] The process initially calculates (at 4905) the number ofrecursions from the track data for the lowest-level slots, pitch pertrack, and size of the design. To do this, the process initiallymultiplies the track data by pitch per track to obtain the dimensions ofthe lowest level slots. It then repeatedly divides the design size by 4,while the resulting value is bigger than the computed dimensions. Witheach division, it increments a level counter by one. The process stopsthe division and the counting, once the resulting value would be smallerthan the computed dimensions.

[0303] Based on the number of levels, the process then computes (at4910) the number of slots. At each level there are 16 children. Hence,the total number of slots is the sum of 16**n, where n varies from 0 tolevel. At this stage, the process also creates a list of all slots.

[0304] Next, the process instantiates (at 4915) slot data structures forthe slots at all the recursion levels. The process then (at 4920) (1)for each net, identifies all the slots that the net traverses, and (2)for each identified slot, instantiates a slot-net data structure tostore the net's configuration in that slot.

[0305]FIG. 50 illustrates a process 5000 that some embodiments use toperform 4920. Specifically, this process is a recursive process thatstarts at the top-level slot and it is performed for each net. Each timethis process is called it receives a slot and a net. As shown in FIG.50, this process 5000 initially computes (at 5005) the bounding box ofthe received slot.

[0306] It then determines (at 5010) whether any pin of the received netis contained in the bounding box of the received slot. If not, theprocess ends. If so, the process creates (at 5015) a slot-net recordthat contains the pin distribution of the net for the current slot. Inaddition, the process recursively calls (at 5020) itself for thereceived net and each child-slot of the current slot. The process 5000then ends.

[0307] After the process 4900 instantiates slot-net data structures tostore the net configurations in the slots, the process 4900 adds (at4925) each circuit module to the table of modules for the slots that itintersects, and then ends. FIG. 51 illustrates a process 5100 that someembodiments use to perform 4925. Specifically, this process is arecursive process that starts at the top-level slot and it is performedfor each circuit module. Each time this process is called, it receives aslot and a circuit module. As shown in FIG. 51, this process 5100initially computes (at 5105) the bounding box of the received slot.

[0308] The process 5100 then computes (at 5110) the bounding box of thereceived circuit module. It then determines (at 5115) whether the twobounding boxes intersect. If not, the process ends. If so, the processadds (at 5120) the module to the list of circuit modules incident to thecurrent slot. In addition, the process recursively calls (at 5125)itself for the received module and each child-slot of the current slot.The process 5100 then ends.

[0309] D. Slot Manager.

[0310]FIG. 52 illustrates a process 5200 performed by the slot manager3925 after the initializer 3915 completes its operation. Initially, theprocess 5200 sets (at 5205) the current level as the top-level slot.Next, the process defines the capacity of the routing paths within theslots of the current level. In some embodiments, the process receives,or can retrieve from a storage structure, the routing path capacitiesfor the first level, and thereafter computes the routing path capacitiesmathematically based on the known geometric relationship between thechild slots and the parent slots. In other embodiments, the process inreal-time calculates the routing path capacities for all the levels.However, some of these embodiments still compute the routing pathcapacities of some recursion levels from those of other levels.

[0311] As mentioned above, some embodiments calculate the capacities ofeach path at a particular recursion level from the size of the edgeorthogonal to the path. For instance, some embodiments calculate thecapacity of each particular path by dividing the size of thecorresponding orthogonal edge (i.e., the size of the edge orthogonal tothe particular path) with the pitch of metal layer corresponding to theparticular path. Some embodiments define the pitch of a metal layer asthe line-to-via pitch. Some embodiments define the line-to-via pitch asthe minimum required distance between interconnect lines on that metallayer, plus ½ the width of the line, plus ½ the width of the viaincluding the metal overlap.

[0312] As mentioned above, in some embodiments, the capacities of thediagonal paths differ from the capacities of the Manhattan paths. Thiscan be modeled by the virtue of the differing size of the edges that areorthogonal to the diagonal and Manhattan paths. It can also be modeledby having the pitch dependent on the type of interconnect line (e.g.,having the pitch for diagonal lines differ from the pitch for theManhattan lines). It can further be modeled by having the pitchdependent on the layer. For instance, in some embodiments, the pitch ofthe −45° metal layer differs from the pitch of the 45° metal layer.

[0313] At 5215, the process sets the current slot as the first slot atthe current level. As further described below, the process 5200 examinesthe slots at the current level in sequence. However, one of ordinaryskill will realize that other embodiments might examine the slots inanother order. For example, some embodiments might examine the mostcongested slots first.

[0314] It then directs (at 5220) the solver to select routes for allnets in the current slot at the current level. Once the solver selectsthe routing paths and stores these paths in the slot-net data structurefield 4515, the slot manager determines whether the current level is thelast recursion level.

[0315] If not, the process directs (at 5230) the propagator to definethe propagation of the selected routes into the child slots of the nextlower recursion level (i.e., into the child slots of the child slots ofthe current slot). For slots that are after the top-level slot, thepropagator also performs a follow-up propagation operation thatpropagates the paths specified by the propagator at the previous routinglevel one level further down. In defining the propagation into the nextlower recursion level, the propagator might modify the net'sconfiguration in the current child slots by adding virtual pins in thechild slots of the current child slots.

[0316] If the process determines that the current slot is at the lastrecursion level (i.e., the current slot is a leaf slot), it directs (at5235) the saver to link the path structures of each net's route in thecurrent leaf slot to their respective net's main data structures. Asmentioned above, the saver also links to the main net data structuresthe path data structures of the propagation paths that the propagatorspecifies for a parent slot of a leaf-level slot (i.e., for agrandparent slot of a Gcell). These propagation paths include paths thatthe propagator identified (1) for the routing paths specified by thesolver and (2) for the routing paths specified by the propagator at theprevious routing level. In this manner, the path data structures linkedto a net's main data structure collectively represent the final routefor the net that the router specifies. In some embodiments, such a routeis a global route for a net.

[0317] From 5230 and 5235, the process transitions to 5240, where itdetermines whether it has examined the last slot at the current level.If not, the process selects (at 5245) another slot at the current leveland returns to 5220 to call the solver for this slot. Otherwise, theprocess determines (at 5250) whether it is at the last recursion level.If not, the process selects the next recursion level, and returns to5210 to specify more detailed routes for the nets at the next lowerrecursion level. When the process determines (at 5250) that it is at thelast recursion level, it ends.

[0318] E. Solver.

[0319] As discussed above, the solver 3930 is responsible for (1)enumerating one or more routes for each net, (2) directing an LP/ILPsolver to select a route for each net, and (3) saving the selectedresults in the slot-net data structures of the current slot. FIG. 53illustrates a process 5300 performed by the solver. In some embodiments,this process starts when the slot manager calls the solver and suppliesit with a slot to route.

[0320] The process 5300 initially predicts (at 5305) congestion ofresources for each path in the slot. One manner for predicting thecongestion of the paths will be described below by reference to FIGS. 54and 55. The process next identifies (at 5310) one or more routes foreach net in the supplied slot. One manner for identifying routes for thenets in the current slot will be described below by reference to FIGS.57-60.

[0321] After identifying one or more routes for each net in the currentslot (i.e., for each net that has a slot-net structure linked to thecurrent slot's data structure), the process 5300 assigns (at 5315) awirelength cost to each retrieved tree by factoring propagation into thenext lower recursion level. In some embodiments, the process uses agreedy technique to account for this propagation. One manner forassigning costs for each retrieved tree will be described below byreference to FIGS. 64.

[0322] Once the solver assigns wirelength costs to the enumeratedpotential routes, the solver formulates (at 5320) the problem for the LPsolver 3945, and the LP solver solves (at 5325) the LP problem. Onemanner for formulating and solving the LP problem will be describedbelow in subsection VI.E.4.

[0323] After 5325, the process 5300 converts (at 5330) the LP solutionto an integer LP (“ILP”) solution. Some embodiments use randomizedrounding to perform this conversion. Randomized rounding is a knowntechnique, and numerous examples of this technique can be found in theliterature, e.g., one such reference is disclosed in RandomizedAlgorithm, by Rajeev Motwani and Prabhakar Raghavan, CambridgeUniversity Press (1995, 1997).

[0324] One example of a randomized rounding process is as follows.First, the process maps the scores returned by the LP solver toprobabilities between 0 and 1. For instance, when the LP solver returnsreal solutions between 0 and 1, a one-to-one mapping exists between thereturned solutions and probabilities between 0 and 1. Second, therounding process generates a random numbers between 0 and 1 for eachnet. Third, the rounding process selects the net's solution that ismapped to the generated random number for the net. Fourth, the roundingprocess measures the quality of the set of selected routes for the netsbased on certain objective function or functions (such as those used bythe LP solver). Fifth, the rounding process iteratively repeats thesecond to fourth operations until the solution space has beensufficiently explored. Sixth, the process selects the set of routes thatresulted in the best quality score.

[0325] Based on the set of routes selected at 5330, the process stores(at 5335) a 42-bit selected route string in each slot-net data structureof the current slot. This 42-bit string specifies the paths in thecurrent slot that the selected route of the net takes. The process thenends.

[0326] 1. Predicting Remaining Path Capacities.

[0327] As mentioned above, the process 5300 predicts (at 5305) thecongestion of path resources in the current slot. In some embodiments,the process specifies the path congestions by estimating the remainingcapacity of each path in the slot. For instance, in some embodiments,the process computes path capacities by initially (1) estimating theunblocked capacity of each path, (2) estimating the use of each path,and (3) subtracting each path's use estimate from its unblocked-capacityestimate. One manner for estimating the unblocked path capacities willbe described below by reference to FIG. 54, while one manner forestimating the path use will be described below by reference to FIG. 55.

[0328] a. Estimating Unblocked Capacity of Each Path.

[0329]FIG. 54 illustrates a process 5400 for estimating the unblockedcapacity of each path in the current slot. Initially, this processallocates (at 5402) a data structure with 42-fields of floating pointnumbers. Each field is for storing the unblocked capacity of one of the42 paths. At 5402, the process also initializes each of path's field inthe data structure to the default capacity value for the path.

[0330] At 5404, the process selects a circuit module in the currentslot's list of circuit module. The process then retrieves (at 5406) thecircuit macro for the selected circuit module. It then selects (at 5408)an obstacle on the circuit macro, and computes (at 5410) the boundingbox of the selected obstacle by using the location of the circuitmodule.

[0331] Next, the process (at 5412) selects one of the 42 paths of thecurrent slot. It then determines (at 5414) whether path selected at 5412is on the same layer as of the obstacle selected at 5408. If theselected path's layer matches the selected obstacle's layer, the processcalculates (at 5422) the bounding box of the selected path. Differentembodiments define bounding boxes for paths differently. For instance,some embodiments define the bounding boxes for both Manhattan andnon-Manhattan paths as rectangular halos about the paths. Under such anapproach, the rectangular halo about a diagonal path is positioneddiagonally with respect to the x-y coordinate axis. Other embodimentsmight define the bounding box of a diagonal path differently. Forinstance, some embodiments might define such a bounding box in terms offour rectangular halos (four bounding boxes) that encompass the fourManhattan edges about the diagonal path (e.g., the bounding box ofdiagonal path 26 would include four boxes that encompass edges E1, E4,E13, and E14).

[0332] The process then calculates (at 5424) the area of the boundingbox of the path. The process next identifies (at 5426) the intersectionof the selected path's bounding box and the selected circuit module'sbounding box, and calculates (at 5428) the area of this intersection.

[0333] It then computes (at 5430) an obstruction factor by dividing thecalculated intersection area by the calculated path area. The processnext multiplies (at 5432) the obstruction factor by the default pathcapacity to produce an estimate of the number of tracks of the pathobstructed by the obstacle. The process then subtracts (at 5434) theresult of this multiplication from the path's current unblockedcapacity, which is stored for the path in the 42-field data structure.The process next transitions to 5416, which is described below.

[0334] The process also transitions to 5416 from 5414 when the selectedpath's layer is not the same as the selected obstacle's layer. At 5416,the process determines whether it has examined all the paths in thecurrent slot. If not, the process returns to 5412 to select another pathof the current slot.

[0335] However, if the selected path is the last path of the currentslot, the process determines (at 5418) whether it has examined all theobstacles of the circuit module selected at 5404. If not, the processtransitions to 5408 to select another obstacle of the selected circuitmodule. Otherwise, the process determines (at 5420) whether it hasexamined all the circuit modules in the current slot. If not, theprocess transitions to 5404 to select another circuit module in thecurrent slot.

[0336] The process 5400 ends when it has examined all the circuitmodules in the current slot. At this stage, the 42-field data structurespecifies the unblocked capacities of the 42 paths in the current slot.Specifically, at this stage, each of the 42fields specifies theunblocked capacity of one of the 42 paths.

[0337] b. Path Use Estimation.

[0338]FIG. 55 illustrates a process 5500 for estimating the use of eachpath in a slot. This process is a recursive process that computes theusage of each path in the current slot in terms of three usagecomponents. One path-usage component represents path usage due to routesbetween current slot's child slots. Another component represents pathcongestion due to routes in the current slot's child slots. The thirdcomponent accounts for the effect of vias on path congestion. One ofordinary skill will understand that other embodiments compute path usagedifferently. For instance, for each net that is in only one child slotof a leaf slot, some embodiments also include a token path-usage valuefor each path incident on the child slot that contains the net.

[0339] The process 5500 starts each time it receives a current slot. Asshown in FIG. 55, the process 5500 initially determines (at 5502)whether the current slot is a leaf slot. If so, the process transitionsto 5512, which will be described below. If not, the process performs5504 to 5510 to compute the usage values due to the routes in thecurrent slot's child slots. Specifically, at 5504, the process selectsone of the child slots of the current slot. The process then estimates(at 5506) the use of each path of each child slot of the current slot.In some embodiments, the process does this by recursively calling itselffor each of the child slots.

[0340] The process next determines (at 5508) whether the child slotselected at 5504 is the last child slot. If not, the process selects (at5504) another child slot and estimates (at 5506) the path-usage in thenewly-selected child slot. Otherwise, the process transitions to 5510 tocalculate the path-usage component due to the congestion in the currentslot's child slots.

[0341] In some embodiments, the process stores the path-usage valuescalculated at 5510 in a data structure (e.g., an array) with 42-fieldsfor storing the usage values for the 42 paths in the current slot. Theprocess receives this data structure with the current slot in someembodiments, while in other embodiments, the process 5500 does notreceive such a data structure but rather initializes the data structurewhen it starts.

[0342] At 5510, the process calculates path-usage component due to thecongestion in the child slots. For instance, the process can define acomponent usage value for path 1 between child slots 1 and 2 in terms ofthe congestion within child slots 1 and 2 via a formula such aspath_1_use =   [(0.75) + (0.25) * (1/(Number  of  recursion  levels − Current  Level))] * (1/2) * (1/2 * (path[1][2] + path[1][5] + path[1][8] + path[1][11]) + 1/3 * (path[1][1] + path[1][4] + path[1][7] + path[1][10]) + 1/6 * (path[1][0] + path[1][3] + path[1][6] + path[1][9]) + 1/2 * (path[2][0] + path[2][3] + path[2][6] + path[2][9]) + 1/3 * (path[2][1] + path[2][4] + path[2][7] + path[2][10]) + 1/6 * (path[2][2] + path[2][5] + path[2][8] + path[2][11])),

[0343] where path[i][j] refers to the usage of path j of child slot i.Similar equations can be used to define analogously the component usagevalues for the other 41 paths in the current slot.

[0344] The above equation only examines in the child slots thehorizontal paths line that are in line with path 1. Specifically, itexamines the path 1's component usage value in terms of all thehorizontal paths (i.e., paths 0-11) of child slots 1 and 2. Thesummation of the usage values in both child slots 1 and 2 is multipliedby ½ to reflect that the capacity of path 1 in the current slot isequally influenced by the capacities of the paths in child slots 1 and2.

[0345] The multipliers ½'s, ⅓'s and ⅙'s are used in the summation forboth child slots 1 and 2 for the following reasons. The objective is toguess how many wires can be pushed through a path. Some of these wireswill terminate immediately after crossing the path, while some willcross the entire width of the slot incident to the path. It is assumedthat there will be a uniform distribution of “endpoints” of the wiresusing the path, such that for propagate 0 of path 1 ¼ will terminate inslot 0 of child slot 2, ¼ will terminate in slot 1 of child slot 2, ¼ inslot 2 of child slot 2 and ¼ in slot 3 of child slot 2 and beyond. Thismeans that ¾ of the wires that use the path 1 will also use path 0 ofchild slot 2, {fraction (2/4)} will use path 1 in child slot 2, and ¼will use path 3 in child slot 2—which gives a ratio of 3:2:1 (or{fraction (3/6)}, {fraction (2/6)}, ⅙) of relative impact of the usagesof these 3 paths on the estimated use of the propagated path.

[0346] Also, the summation of usage values in child slots 1 and 2 ismultiplied by [(0.75)+(0.25)*(1/(Number of recursion levels−CurrentLevel))]. This multiplication is based on an assumption that most netsroutes' include a single path connecting two real pins. If a uniformdistribution of pins is assumed in the grandchild slots, then ¾^(th) ofthe routes using a given path will see congestion from the nets whollycontained in each child slot. As the router moves down the hierarchy, agreater percentage of the routes traverse entire grandchild slots, andhence more nets will see the entire congestion in the grandchild slots,which thereby justifies increasing the multiplier to 1.

[0347] From 5502 or 5510, the process transitions to 5512. The processperforms operations 5512 to 5528 to compute the component usage valuesdue to the routes between the current slot's child slots. The processcomputes these component values in terms of theprobabilistic-Steiner-tree contributions described above.

[0348] At 5512, the process selects a net. The process then retrieves(at 5514) the selected net's pin configuration in the current slot. Itnext identifies (at 5516) the probabilistic Steiner-tree values for theretrieved pin configuration. As mentioned above, FIG. 28 illustrates theprobabilistic Steiner-tree values for the pin configuration of net 505illustrated in FIG. 5. Some embodiments precompute the probabilisticSteiner-tree values, as described above by reference to FIG. 26. Otherembodiments, however, generate these values during the path-useestimation process 5500.

[0349] The process then adds (at 5520) each path's probability values tothe path's usage value in the 42-field data structure. Next, the processperforms 5522-5530 to compute the token usage value to account for theeffect of vias for the net on the path congestion. When a net has one ormore pins in the child slots of the current slot, these pins aretypically on the lower metal layers. Accordingly, vias will need to beadded to connect some of these pins to the specified paths for the net.The addition of each via, in turn, will increase the path congestion.

[0350] At 5522, the process selects a child slot that contains one ofthe pins of the net selected at 5512. This net has one or more routes,with each route using one or more paths. Hence, at 5524, the processselects one of the route paths incident on the child slot selected at5522. It then increments (at 5526) the selected path's usage value by atoken amount. In some embodiments, the token amount is 0.5/(Number ofRecursion levels−Current Recursion Level).

[0351] The process then determines (at 5528) whether it has examined allthe paths (of all the trees of the selected net) that are incident onthe selected child slot. If not, the process selects (at 5524) anotherpath incident on the selected child slot and then increments (at 5526)the selected path's usage value by the token amount.

[0352] On the other hand, if the process determines (at 5528) that ithas examined the last path incident on the selected child slot, theprocess determines (at 5530) whether it has examined the last child slotwith a pin for the selected net. If not, the process returns to 5522 toselect another child slot that has a pin for the selected net.Otherwise, the process determines (at 5532) whether it has examined thelast net in the current slot. If not, the process transitions back to5512 to select another net and to perform the subsequent operations forthis net. The process ends when it determines that it has examined thelast net at 5532.

[0353] 2. Identifying Routes for Each Net in the Current Slot

[0354] After estimating (at 5305) the remaining capacity of each path inthe current slot, the process identifies (at 5310) one or more routesfor each net in the current slot. The embodiments described belowinitially use each net's configuration with respect to the current slotto identify one or more routes for each net.

[0355] These embodiments then generate fake configurations for some orall nets, and identify additional routes for the nets based on thegenerated fake configurations. Some embodiments use two differentapproaches to generate fake configurations for a net. One approach addsfake pins to a net's configuration. This approach is described below byreference to FIGS. 56-58. The second approach breaks a net'sconfiguration into two or more configurations and adds fake pins to thenew configurations. This approach is described below by reference toFIGS. 60 and 59.

[0356] a. Identifying Routes for Each Net Configuration and GeneratingDetour Possibilities by Adding Fake Pins to the Net Configurations.

[0357]FIG. 56 illustrates a process 5600 for identifying routes for eachnet configuration and generating detour possibilities by adding fakepins to the net configurations. As shown in this figure, the process5600 initially selects (at 5602) a net in the current slot. At 5604, theprocess (1) uses the net's configuration in the current slot to identifyroutes for the selected net, and (2) stores the identified routes forthe selected net in a variable in the solver. The process 5600identifies trees for the particular net configurations based on one ofthe approaches described above in Section V.

[0358] After identifying and storing the routes based on the selectednet's configuration, the process performs some or all of the operations5606 to 5668 to determine whether it needs to generate detour routes forthe selected net, and if so, to generate these detour routes by addingone or two fake pins.

[0359] The process 5600 generates detour routes for a selected net whereall the optimal routes identified for the net at 5604 use one or morepaths that are “at risk”. A path is “at risk” if the estimatedcongestion (path-use plus blockages) is near or over the path'scapacity. Some embodiments determine whether a path is “at risk” bydetermining whether the path's remaining capacity (which was computed at5305) is less than a threshold amount. Sub-optimal routes can begenerated for a varying number of nets by varying the threshold at whicha path is defined to be “at risk.”

[0360]FIGS. 57 and 58 provide illustrative examples of how sub-optimaldetour routes are generated by adding one or two fake pinconfigurations. Specifically, FIG. 57 illustrates a sub-optimal route5725 for a net that has two pins 5710 and 5715 in child slots 8 and 11.This sub-optimal route is generated by adding a fake pin 5705. Thissub-optimal route 5725 avoids horizontal path P7 (between child slots 9and 10), which is congested due to obstacle 5720. Adding virtual pin5705 to the pin distribution of the net results in a new pinconfiguration. The optimal route for this new pin configuration providesa sub-optimal route for the original net having pins 5710 and 5715. Thissub-optimal route does not use “at risk” path 7.

[0361]FIG. 58 illustrates an example where two fake pins 5805 and 5810have been added to a net that has pins 5815 and 5820. These two fakepins generate a sub-optimal route 5830 that avoids paths 7 and 33 thatare blocked and congested by obstacle 5825.

[0362] The process 5600 of FIG. 56 generates detour routes by (1) addingone and then two fake pins to the net configurations, and (2)identifying the best optimal routes for the resulting pinconfigurations. Even though this process at most adds two fake pins to anet's configuration, one of ordinary skill will realize that otherembodiments add more fake pins to the net configurations in order toidentify useful detour routes for some or all the nets.

[0363] At 5606, the process selects one of the routes identified at5604. It then identifies (at 5608) one of the paths of the routeselected at 5606. At 5610, the process determines whether the remainingcapacity of the path selected at 5608 is less than the thresholdcapacity value. The remaining capacity of the path was computed at 5305.

[0364] If the process determines (at 5610) that the path's remainingcapacity is less than its threshold capacity value, the process flags(at 5612) the route selected at 5606 as unusable, and then transitionsto 5616, which will be described below. If not, the process determines(at 5614) whether it has examined all the paths of the selected route.If it has not examined all the paths, it transitions back to 5608 toselect another path of the selected route. Otherwise, the processtransitions to 5616.

[0365] At 5616, the process determines whether it has examined all theroutes identified at 5604. If not, the process transitions back to 5606to select another of the identified routes for the selected net.Otherwise, the process determines (at 5618) whether it marked all theidentified routes for the selected net as unusable. If not, the net hasone or more routes that do not use any “at risk” paths, and hence theprocess does not generate fake configurations for this net. The processthen determines (at 5620) whether it has examined all the nets in thecurrent slot. If it has examined all the nets, the process ends.Otherwise, the process transitions from 5620 to 5602 to select anothernet in the current slot.

[0366] If the process determines (at 5618) that all the identifiedroutes for the selected net are unusable, the process selects (at 5622)a child slot of the current slot. For the selected net, the process thengenerates (at 5624) a fake pin configuration that indicates the net hasa pin in the child slot selected at 5622. In some embodiments, theprocess generates this fake pin configuration by duplicating theselected net's actual pin configuration in the current slot, andensuring that the duplicated pin configuration indicates a pin for thechild slot selected at 5622. In the example illustrated in FIG. 57, theprocess generates a fake configuration 0000100101000000 by adding a “1”for the 6

[0367] th child slot to the original configuration 0000100100000000.

[0368] The process then identifies (at 5626) routes for the fake pinconfiguration generated at 5624. The process can identify these routesfor the pin configuration by using any one of the methods discussedabove in Section V. Next, the process selects (at 5628) one of theroutes identified at 5626. The process then determines (at 5630) whetherthe selected route is usable. The process makes this determination byperforming usability-checking operations similar to 5608-5616, whichwere described above. If process determines (at 5630) that the routeselected at 5628 is not usable, the process transitions to 5634, whichwill be described below. If the selected route is usable, the processrecords (at 5632) for the generated fake configuration the selectedroute's cost, and then transitions to 5634.

[0369] At 5634, the process determines whether it has examined all theroutes identified at 5626 for the generated fake configuration. If not,the process transitions back to 5628 to select another identified route.Otherwise, the process determines (at 5636) whether it has generated afake pin configuration for all the child slots of the current slot. Ifnot, the process returns to 5622 to select another child slot.

[0370] Otherwise, the process identifies (at 5638) the fake pinconfiguration, if any, that resulted in the useable route with the bestcost recorded at 5632. If a configuration is identified at 5638, theprocess then identifies (at 5640) all the useable routes for the pinconfiguration identified at 5638, and adds these routes to the set ofpossible routing solutions for the current net.

[0371] Next, the process performs 5642-5668 to generate routes for fakeconfigurations that contain up to 2 fake pins. Specifically, at 5642,the process selects a child slot of the current slot. The processduplicates (at 5644) the selected net's actual pin configuration in thecurrent slot. It then ensures (at 5646) that the duplicated pinconfiguration indicates a pin for the child slot selected at 5642. Inthe example illustrated in FIG. 58, the process generates an initialfake configuration 0000100101000000 by adding a “1” for the 6^(th) childslot to the original configuration 0000100100000000.

[0372] At 5648, the process then selects a child slot other than the oneselected at 5642. It then generates (at 5650) a fake pin configurationthat indicates the net has a pin in the child slot selected at 5648. Insome embodiments, the process generates this fake pin configuration byduplicating the pin configuration identified at 5646, and ensuring thatthe duplicated pin configuration indicates a pin for the child slotselected at 5648. In the example illustrated in FIG. 58, the processgenerates a final fake configuration 0000100101100000 by adding a “1”for the 5^(th) child slot to the initial fake configuration0000100101000000.

[0373] The process then identifies (at 5652) routes for the fake pinconfiguration generated at 5650. As at 5604 and 5626, the process canidentify these routes for the generated pin configuration by using anyone of the methods discussed above in Section V.

[0374] Next, the process selects (at 5654) one of the routes identifiedat 5652. The process then determines (at 5656) whether the selectedroute is usable. The process makes this determination by performingusability-checking operations similar to 5608-5616, which were describedabove. If the route selected at 5654 is not usable, the processtransitions to 5660, which will be described below. If it is usable, theprocess records (at 5658) for the generated fake configuration theselected route's cost, and then transitions to 5660.

[0375] At 5660, the process determines whether it has examined all theroutes identified at 5652 for the fake pin configuration generated at5650. If not, the process transitions back to 5654 to select anotheridentified route. Otherwise, the process determines (at 5662) whether ithas generated a fake pin configuration for all the child slots otherthan the child slot selected at 5642. If not, the process returns to5648 to select another child slot other than the one selected at 5642.

[0376] Otherwise, the process determines (at 5664) whether it hasexamined all the child slots as potential first fake pins. If not, theprocess returns to 5642 to select another child slot. When the processdetermines (at 5664) that it has examined all the child slots aspotential first fake pins, the process identifies (at 5666) the fake pinconfiguration, if any, that resulted in the useable route with the bestcost recorded at 5658. If a configuration was identified at 5666, theprocess then identifies (at 5668) all the useable routes for the pinconfiguration identified at 5666, and adds these routes to the set ofpossible routing solutions for the current net.

[0377] From 5668, the process transitions to 5620. At 5620, the processdetermines whether it has examined all the nets in the current slot. Ifit has examined all the nets, the process ends. Otherwise, the processtransitions from 5620 to 5602 to select another net in the current slot.

[0378] b. Breaking Net Configurations into Smaller Pin Configurationsand Adding Fake Pins to the Smaller Pin Configurations

[0379] In some instance, simply adding fake pins to a net configurationdoes not result in the best sub-optimal route. Such a route cansometimes be generated by (1) generating two or more pin configurationsfrom a net's pin configuration, (2) identifying fake configurations forthe generated pin configurations, (3) identifying routes for theconfigurations, and (4) combining the resulting routes to find one ormore sub-optimal routes. Such an approach is especially useful insituations where the congested path is between 2 adjacent “real” pins.

[0380]FIG. 59 illustrates an example of such an approach. In thisexample, a net has two pins 5905 and 5910. If a fake pin 5915 is addedto the original net configuration, the route for the resulting netconfiguration would use paths P6 and P21. However, such a route wouldnot be useable since obstacle 5920 completely obstructs path P6. A moreideal route that uses paths 21 and 36 can be obtained by (1) splittingthe pin configuration into two, one that contains pin 5905 and one thatcontains pin 5910, and (2) adding the fake pins 5915 to both of theresulting pin configurations.

[0381]FIG. 60 illustrates a process 6000 that identifies additionalroutes for a net configuration. This process generates two pinconfigurations from the net's pin configuration, identifies fakeconfigurations for both the generated pin configurations, identifiesroutes for the fake configurations, and combines the resulting routes.Some embodiments perform this process for each net in the current slot,while other embodiments only perform this process for some of the nets,such as those for which the process 5600 was not able to find useableroutes.

[0382] As shown in FIG. 60, the process 6000 starts by identifying (at6002) one or more routes for a net in the current slot. The process canidentify these routes based on the net's pin configuration by using anyone of the methods discussed above in Section V. The process thenidentifies (at 6004) the child slot that has the most number of paths ofthe identified routes incident upon it. The process then defines (at6006) two bitsets, bset1 and bset2. In some embodiments, each bitset has16-bits, all of which are initially defined to be zero.

[0383] Next, at 6008, the process sets to 1 the first bitset's bitcorresponding to the child slot identified at 6004. At 6008, the processalso sets to 1 the second bitset's bit or bits that correspond to theremaining child slots that contain pins of the current net. At 6010, theprocess initializes two variables, which are the Best_Length variableused to measure the best length of a number of solutions, and theBest_Detour variable used to identify the solution that resulted in thebest length. The Best_Length variable is initialized to a large value,while the Best_Detour variable is initialized to null.

[0384] At 6012, the process selects one of the current slot's childslots. It then determines (at 6014) whether the current net's pinconfiguration has the selected child slot's corresponding bit set to 1(i.e., whether the current net has a pin in the selected child slot). Ifso, the process determines (at 6016) whether it has tried to generatefake configurations for each child slot of the current slot. If it hasnot examined all the child slots of the current slot, the processreturns to 6012 to select another child slot. If it has, the processtransitions to 6040, which will be described below.

[0385] If the process determines (at 6014) that the bit corresponding tothe selected child slot is not set to 1 in the current net's pinconfiguration, the process generates (at 6018) two duplicated bitsets,biset1 c and bitset2 c, from the two bitsets, bitset1 and bitset2. Ineach duplicate bitset, the process then sets (at 6020) to 1 the bit thatcorresponds to the selected child slot.

[0386] Next, the process identifies (at 6022) one or more routes foreach of the duplicated, modified bitsets. The process can identify theseroutes for the duplicated, modified bitsets by using any one of themethods discussed above in Section V. The process then selects (at 6024)one of the routes for the first duplicated, modified bitset (i.e., forbitset1 c), and selects (at 6026) one of the routes for the secondduplicated, modified bitset (i.e., for bitset2 c). At 6028, the processthen determines whether the two selected routes overlap (i.e., whetherthe two solutions share one or more routing paths between the childslots). If so, the process transitions to 6036, which will be describedbelow.

[0387] If not, the process calculates (at 6030) the length of a routethat would result by combining the two routes selected at 6024 and 6026.In some embodiments, the process calculates this length by summing thelengths of the two routes selected at 6024 and 6026. At 6032, theprocess determines whether the length calculated at 6030 is less thanthe current Best_Length. If not, the process transitions to 6036, whichwill be described below.

[0388] When the length calculated at 6030 is less than the currentBest_Length, the two routes selected at 6024 and 6026 represent the bestsolution that the process 6000 has seen thus far. Accordingly, at 6034,the process defines the Best_Length equal to the length computed at6030. At 6034, the process also generates a new route as the combinationof the two routes selected at 6024 and 6026, and defines the Best_Detouras the generated new route. From 6034, the process transitions to 6036.

[0389] At 6036, the process determines whether it has examined all theroutes identified (at 6022) for the second duplicated, modified bitset(bitset2 c) with the route that was selected at 6024 for the firstduplicated, modified bitset (bitset1 c). If not, the process returns to6026 to select another route for the second bitset2 c. Otherwise, at6038, the process determines whether it has examined all the routesidentified (at 6022) for the first duplicated, modified bitset (bitset1c). If not, the process returns to 6024 to select another route for thefirst bitset1 c to examine.

[0390] When the process determines (at 6038) that it has examined allthe identified routes for the first duplicated, modified bitset (bitset1c), it determines (at 6016) whether it has tried to generate fakeconfigurations for each child slot of the current slot. If it has, theprocess (at 6040) adds the solution (if any) that the process identifiedas the Best_Detour to the current net's solution pool, and then ends. Onthe other hand, if the process has not generated fake configuration foreach child slot, the process returns to 6012 to select another childslot.

[0391] 3. Assigning Costs for the Potential Routes

[0392] After identifying (at 5310) one or more routes for each net inthe current slot, the process 5300 assigns (at 5315) a wirelength costto each retrieved tree by factoring propagation into the next lowerrecursion level. In some embodiments, the cost of each route includesthe following three component costs: (1) the wirelength cost of theroute's path or paths that connect the child slots of the current slot,(2) the path propagation cost of the route's path or paths into thechild slots, and (3) the cost in each child slot after selecting thepath propagations.

[0393]FIG. 64 illustrates a process for calculating the cost of eachroute in terms of these three component costs. Before explaining thisprocess, the conceptual framework used by this process is explained byreference to FIGS. 61-63. These figures illustrate how some embodimentsmodel propagation of a higher-level route into lower level child slots.Specifically, FIG. 61 illustrates that any horizontal or vertical pathbetween the current slot's child slots can propagate down into the slotsof the child slots along one of 10 propagation paths.

[0394]FIGS. 62 and 63 illustrate two different ways for modeling thepropagation of a 45° diagonal path into the lower level child slots. Themodel that FIG. 62 illustrates provides 7 propagation possibilities fora 45° diagonal path. On the other hand, the model illustrated in FIG. 63provides for 19 propagations for a 45° diagonal path between twodiagonally-positioned child slots 6305 and 6310. This is because themodel illustrated in FIG. 63 only specifies the path propagations alongthe edges of child slots 6305 and 6310, which thereby allows a pathpropagation along an edge of one of the child slots to pair up withanyone of three path propagations along the corresponding edge of theother child slot. For instance, as shown in FIG. 63, the propagation6315 along edge 6320 can be paired up with anyone of the propagations6325, 6330, and 6335 along edge 6340. Under this approach, a diagonalpath between slots 6305 and 6310 can have 19 propagations, where (1) 9of the propagations are defined by pairing up three path propagationsalong edge 6320 with three path propagations along edge 6340, (2) 9 ofthe propagations are defined by pairing up three path propagations alongedge 6345 with three path propagations along edge 6350, and (3) 1 of thepropagation 6355 is defined between the top left hand corner of slot6310 and the bottom right hand corner of slot 6305.

[0395] As shown in FIG. 64, the process 6400 initially selects (at 6402)a net. It then selects (at 6404) one of the routes identified (at 5310)for the selected net. The process next initializes (at 6406) theselected-solution's Total_Cost to 0. It then needs to identify the pathcost of the route selected at 6404. In some embodiments, one of theLUT's 3965 stores pre-tabulated wirelength costs for each netconfiguration. In these embodiments, the pre-tabulated wirelength costof a selected tree can be retrieved from the LUT when the tree is addedto the solution set of the net. Alternatively, the process 6400 can usethe pin configuration that resulted in the identification of theselected tree for the current net to retrieve the tree's pre-tabulatedwirelength cost from the LUT.

[0396] In the embodiments described below, however, the process 6400computes the path cost of the selected route by performing 6408-6412.Specifically, at 6408, the process selects a path of the selected route.It then increments (at 6410) the selected-solution's Total_Cost by thecost of the path selected at 6408. Some embodiments define a path's costpurely based on the relative length of the path compared to other paths.For instance, some embodiments assign a path-cost of 1 for eachhorizontal or vertical path between slots (e.g., for each path P0-P23 inFIG. 12) and a path-cost of 1.4 for each diagonal path between slots(e.g., for each paths P024-P041). Other embodiments assign a path costof 5 for each horizontal or vertical path between slots (e.g., for eachpath P0-P23 in FIG. 12) and a path cost of 7 for each diagonal pathbetween slots (e.g., for each paths P024-P041).

[0397] Other embodiments might define the path-costs not purely based onthe relative path lengths. To achieve a certain objective (e.g.,encourage use of the lower-layer wiring or discourage use of vias), someembodiments might cost the paths that traverse the higher levels moreexpensive than their relative length cost compared to the lower-layerwiring. For example, some embodiments might assign a path-cost of 1 foreach horizontal or vertical path between slots and a path-cost greaterthan 1.4 for each diagonal path between slots.

[0398] At 6412, the process determines whether it has examined all theselected route's paths. If not, the process returns to 6408 to selectanother path for costing. When the process determines (at 6412) that ithas examined all the selected route's paths, it determines (at 6414)whether the current slot is a leaf-level slot. If so, the processtransitions to 6428, which will be described below.

[0399] If the process determines that the current slot is not aleaf-level slot, the process accounts for the wirelength cost in thechild slots. The process uses a greedy technique to factor thepropagation of the selected route into the child slots. Specifically,the process orders (at 6416) each of the selected tree's paths by thenumber of anchors of that path. Some embodiments define an anchor as apin in either child slot upon which the path is incident. In theseembodiments, a path has at most two anchors. Other embodiments mightdefine an anchor as the number of pins in the slots of a child slot;under such an approach, a path can have up to 32 anchors, when it has 16pins in the 16 slots of each child slot. In some embodiments, theprocess sorts the paths in the order of descending number of anchors(i.e., places the paths with the most number of anchors first on itssorted list).

[0400] Next, at 6418, the process selects a path according to the sortedorder (i.e., selects the paths on the sorted list in a top-to-bottommanner). For the selected path, the process selects (at 6420) one of thepropagation possibilities. In other words, at this stage, the processselects one of the ways that the selected path can propagate between thetwo slots that it is incident upon.

[0401] As mentioned above by reference to FIG. 61, some embodiments usea propagation model that provides 10 potential propagations for ahorizontal or vertical path between the current slot's child slots.Also, some embodiments use a propagation model that provides 7 potentialpropagations for a 45° path as shown in FIG. 62, while other embodimentsuse a model that provides 19 potential propagations for a 45° path asshown in FIG. 63.

[0402] In some embodiments, the process selects (at 6420) the optimalpropagation for the selected path. To select the optimal propagation,the process examines each propagation possibility, adds virtual pinswhen necessary, costs each propagation possibility, and chooses thepropagation possibility that results in the lowest cost propagation androutes. As described below by reference to FIG. 65, each propagationpossibility might specify one or two propagation paths that traverseinto two or three child slots. Accordingly, the cost of each propagationpossibility includes the cost of the propagation path(s) plus therouting cost of the pin configurations in the two or three child slotsthat the propagation possibility traverses.

[0403]FIG. 65 illustrates one example of a propagation possibility of apath. This figure illustrates a net that has actual pins 6525 in slots 0and 9. The selected route for this net uses paths P17 and P24 thattraverse child slot 5 to connect child slots 0 and 9. FIG. 65illustrates that the path P24 is propagated along paths 6510 and 6515into three child slots, (i.e., child slots 0, 1, and 5), while path P17is propagated along path 6520 into two child slots 5 and 9. Thepropagation path 6510 is between the child slot 7 of the current slot'schild slot 0 and the child slot 8 of the current slot's child slot 1.The propagation path 6515 is between the child slot 13 of the currentslot's child slot 1 and the child slot 2 of the current slot's childslot 5. The propagation path 6520 is between the child slot 14 of thecurrent slot's child slot 5 and the child slot 2 of the current slot'schild slot 9. FIG. 65 illustrates five virtual pins 6505 that have beenadded to the slots of child slots 1, 5, and 9.

[0404] At 6420, the process also (1) computes the delta cost between thecost of the path(s) of the propagation possibility identified at 6420and the cost of the path selected at 6418, and then (2) increments theTotal_Cost with this delta. For instance, in some embodiments, the deltacost is two when the selected path is a Manhattan path with a cost of 5and the identified propagation is a diagonal path with a cost of 7.Also, in some embodiments, the delta cost is seven when the selectedpath is a diagonal path with a cost of 7, and the identified propagationincludes two diagonal paths, each with a cost of 7. In some embodiments,the delta cost is 0 when the selected path and its identifiedpropagation are both Manhattan paths.

[0405] For the propagation selected at 6420, the process, if necessary,temporarily stores (at 6422) virtual pins in the pin configurationrecords of the incident child slots (i.e., the child slots upon whichthe propagation identified at 6430 is incident). The process uses thesetemporarily stored virtual pins in computing the costs of thepropagations of other paths (if any) of the selected route.

[0406] The process next determines (at 6424) whether it has examined thelast path of the selected route. If not, the process returns to 6418 toselect the next path on the sorted path list. It then identifies (at6420) the best propagation for this newly-selected path and temporarilysets (at 6422) any necessary virtual pins.

[0407] When the process determines (at 6424) that it has examined allthe paths of the selected route, the process increments (at 6426) theselected tree's Total_Cost with the cost of the routes in each childslot that has a pin set for the net. At 6426, the process also storesthe selected route's Total_Cost. The solving process 5300 uses eachtree's Total_Cost to formulate its LP problem.

[0408] At 6428, the process determines whether it has examined all theroutes for the net selected at 6402. If not, the process transitions to6404 to cost the next route for this net. When the process determines(at 6428) that it has examined all the routes for the net selected at6402, the process determines (at 6430) whether it has examined all thenets in the current slot. If not, the process transitions back to 6402to select another net in the current slot and to perform the subsequentoperations for costing the routes for this net. When the processdetermines (at 6430) that it has examined all the nets in the currentslot, the costing process 6400 ends.

[0409] One of ordinary skill will realize that other embodiments computethe wirelength cost of a route differently than the process 6400. Forinstance, the process 6400 computes this cost by computing thewirelength cost at the current recursion level and the one below it.Other embodiments, on the other hand, might compute this cost bycomputing the wirelength cost from the current recursion level all theway down to the leaf-level slots. To do this, some embodiments use arecursive process that computes the path-cost at the current recursionlevel and then recursively call the cost-computing process to computethe wirelength cost for each child slot that is not a Gcell (i.e., foreach child slot that is not a child of a leaf-level slot).

[0410] 4. LP Problem Formulation and Solver

[0411] After the solver assigns wirelength costs to the routes of thenets in the current slot, the solver formulates (at 5320) an LP problemfor the LP solver 3945, which then solves (at 5325) this LP problem. Thebasic variables in the LP-problem formulation are the routes for thenets in the current slot. Each tree is represented in the followingformat: “xN_C”, where “x” is a character constant, “N” is the netnumber, “_” is a character constant, and “C” is a number that identifiesthe tree in the list of trees for the net. For instance, X26_14represents the 14^(th) tree of net 26.

[0412] In some embodiments, each LP solution examined by the LP solverincludes a real number value for each tree variable xN_C. The task ofthe LP solver is to identify an LP solution that minimizes one or moreobjective functions while satisfying a number of constraints.Specifically, a solution is a viable LP solution only if it satisfiesthe constraint or constraints of the specified LP problem. The LPsolver's task is to identify a viable LP solution (i.e., a solution thatsatisfies the specified constraints) that minimizes the objectivefunction. In other words, from a set of solutions, the LP solveridentifies the viable LP solution that produces the bestobjective-function value.

[0413] Some embodiments use as the LP solver 3945 the “SoPlex” solver,which has been implemented by Roland Wunderling as a part of a Ph.D.thesis entitled “Paralleler und Objektorientierter Simplex-Algorithmus”(in German). Information about this solver is available at the followingwebsite: http://www.zib.de/Optimization/Software/Soplex/.

[0414] a. Objective Function

[0415] Different embodiments use different objective functions. Forinstance, some embodiments might use an objective function that (1)minimizes total length; (2) maximizes minimum slack across all 42 paths;(3) maximizes total slack; and (4) minimizes maximum usage of anyindividual path. The embodiments described below, however, try to findan LP solution that minimizes the following objective function:

Objective FunctionA*Total_WireLength+B*Total_Via_Number−C*Min_Slack.  (F)

[0416] In the equation (F), A, B, and C are weighting factors. Also, theTotal_WireLength is: $\begin{matrix}{{{Total\_ WireLength} = {\sum\limits_{Tree}\quad {\left( {{Length}\quad {Cost}\quad {of}\quad {the}\quad {Tree}} \right)*\left( {{Variable}\quad {for}\quad {the}\quad {Tree}} \right)}}},} & (G)\end{matrix}$

[0417] and the Total_Via_Number is: $\begin{matrix}{{{Total\_ Via}{\_ Number}} = {\sum\limits_{Tree}\quad {\left( {{Number}\quad {of}\quad {Vias}\quad {for}\quad {the}\quad {Tree}} \right)*\left( {{Conversion}\quad {Factor}} \right)*{\left( {{Variable}\quad {for}\quad {the}\quad {Tree}} \right).}}}} & (H)\end{matrix}$

[0418] The notion of minimum slack is used in two instances in the LPformulation. First, the variable Min_Slack is used as a component of theobjective function for quantifying the congestion (i.e., serves as anindicia of the congestion in the objective function). Second, theconstant minSlack is used to specify a minimum slack that can betolerated across all 42 paths.

[0419] A path's slack is the remaining capacity of a path afteraccounting for all congestion (i.e., blockages and wireflow) for aparticular solution. A negative slack signifies that a path is overcongested. Minimizing the objective function minimizes the negativeminimum slack, which, in turn, maximizes the minimum slack. One ofordinary skill will realize that other embodiments might use othercongestion indicia in the objective function.

[0420] In some embodiments, the weighting factors A and B for theTotal_WireLength and Total_Via_Number are set equal to each other, andboth these factors are larger than the weighting factor C for the Min_Slack. In other words, these embodiments weight the objective functiontowards the wirelength and the via count, so that the Min_Slackcomponent makes a difference only when the wirelength and via countcomponents cannot distinguish two LP solutions. These embodiments usethe wirelength and via count components in the objective function toselect solutions that result in smaller total wirelengths and viacounts.

[0421] Other embodiments weight the objective function differently. Forinstance, the LP-formulation listed below in subsection VI.E.4.c weightsthe objective function towards the wirelength and the via countparameters only for the first attempt of the LP solver to solve theformulated LP problem. If the LP solver cannot solve the formulatedproblem in its first iteration (i.e., if it cannot find a solution thatmeets the constraints), the LP problem is reformulated so that theMin_Slack component becomes the primary component in the objectivefunction. Specifically, for the first iteration of the LP solver, theformulation below sets the weighting factors A and B for theTotal_WireLength and Total_Via_Number equal to each other, and boththese factors have a larger magnitude than the weighting factor C forthe Min_Slack. If the LP solver does not solve the LP problem in itsfirst iteration, the weighting factors A, B, and/or C are changed sothat the objective function's primary parameter is Min_Slack.

[0422] As illustrated by equation (G) above, the Total_WireLength iscomputed by using the cost of each tree, which was computed at 5315.Also, as illustrated by equation (H) above, the Total_Via_Number iscomputed by using the number of vias for each tree. A via is aconnection that is needed to connect two parts of a route that are ontwo adjacent metal layers.

[0423] For the octagonal wiring model illustrated in FIG. 3, a one toone mapping exists between layers and routing directions. This mappingcan be used to easily compute the number of vias for each route. Hence,the number of vias can be counted by traversing the route, andidentifying the number of vias necessary to accommodate (1) the changesin direction of the route and (2) the difference, if any, between thelayers of pins and paths in the child slots.

[0424] To account for the layer difference between a pin in a child slotand path incident upon the child slot, the pin's layer needs to beidentified. Different embodiments identify the layer of an actual pin(i.e., a non-virtual pin) in different ways. For instance, someembodiments assume that all actual pins are on layer 2. Otherembodiments identify the actual layer of a pin; as mentioned above, someembodiments store a pin's layer as part of the pin's location that isstored in the pin data structure 4300, while other embodiments store thepin's layer as part of a pin macro that is stored with the circuit macroreferenced by the slot-data structure. Some embodiments define the layerof a virtual pin (i.e., a pin that is set to account for a propagationof a route into a lower-level child slot) to coincide with the layer ofthe propagated path for which the virtual pin was set.

[0425]FIGS. 66 and 67 present two examples that conceptually illustratesone manner of counting the number of vias. In these examples, the wiringmodel is as follows: layer 2 is vertical, layer 3 is horizontal, layer 4is +45°, and layer 5 is −45°. FIGS. 66 and 67 illustrate two routes forconnecting child slots 0 and 9. In both these figures, child slot 0includes a virtual pin 6605 that was set to account for the propagationof a +45° path into slot 6600. Accordingly, the virtual pin 6605 is saidto be on the fourth metal layer (i.e., the metal layer for the +45°wiring). Also, child slot 9 includes an actual pin, which is on layer 2.

[0426] Route 6610 of FIG. 66 needs two vias. This number can be countedby starting at child slot 0. This slot has one virtual pin in the fourthlayer. Path P24 is incident on this slot, and it traverses the fourthlayer. Hence, no via is necessary to account for the difference betweenthe layers of pin 6605 and path P24. Path P24 is also incident on childslot 5. Child slot 5 has no pin, but it has path P17 incident upon it.As path P24 is on the fourth layer and path P17 is on the second layer,two vias are needed to account for the changes of path direction inchild slot 5. Path P17 is also incident upon child slot 9. Slot 9 has noother paths incident upon it, but it has an actual pin 6615 on layer 2.Given that path P17 and pin 6615 are both on layer 2, no via isnecessary to connect pin 6615 and path P17.

[0427] Route 6705 of FIG. 67 needs six vias. This number can be countedby starting at child slot 0. This slot has one virtual pin in the fourthlayer. Path P12 is incident on this slot, and it traverses the secondlayer. Hence, two vias are necessary to account for the differencebetween the layers of pin 6605 and path P12. Path P12 is also incidenton child slot 4. Child slot 4 has no pin, but it has path P30 incidentupon it. As path P30 is on the fourth layer and path 12 is on the secondlayer, two vias are needed to account for the changes of path directionin child slot 4. Path 30 is also incident upon child slot 9. Slot 9 hasno other paths incident upon it, but it has an actual pin 6615 on layer2. Hence, two vias are necessary to account for the difference betweenthe layers of pin 6615 and path P30. In sun, six vias are necessary forroute 6705. As it can be seen from the examples illustrated in FIGS. 66and 67, the number of vias provides a useful indicia for distinguishingtwo routes that have equal lengths.

[0428] Also, some embodiments compute the number of vias for each routeduring the LP formulation 5320. One such embodiment is illustrated bythe LP formulation in subsection VI.E.4.c. Other embodiments, however,count the number of vias for each route before 5320, just as theycompute the wirelength cost of a route before 5320 at 5315.Alternatively, some embodiments might compute the wirelength cost ofeach route during the LP formulation 5320.

[0429] FIGS. 68-70 illustrate three processes that work together tocompute the number of vias in a route. Process 6800 of FIG. 68 startswhenever it is called to compute the via count for a tree. This processinitially initializes (at 6805) all slots (of the slot currently beingsolved) as slots that have not been visited.

[0430] At 6810, the process 6800 then selects a slot that has only onepath of the route incident upon it, and defines this slot as theCurrent_Slot. It then calls (at 6815) process 6900 of FIG. 69 andsupplies this process with the Current_Slot. The value returned byprocess 6900 is the total number of vias. After calling process 6900,the process 6800 ends.

[0431] The process 6900 is a recursive process. It initially computes(at 6905) the number of vias in the Current_Slot. In some embodiments,the process 6900 computes this number by calling the process 7000 ofFIG. 70. The process 7000 starts by identifying (at 7005) all theroute's paths that are incident on the Current_Slot. It then identifies(at 7010) all the actual and virtual pins in the Current_Slot.

[0432] The process 7000 next identifies (at 7015) the layer of eachroute path identified at 7005 and each pin identified at 7010. Eachpath's layer can be easily determined as there is a one-to-one mappingbetween the path's direction type and its layer, when the octagonalwiring model of FIG. 3 is used. For instance, some embodiments mapvertical paths to layer 2, horizontal paths to layer 3, +45° paths tolayer 4, and −45° paths to layer 5. Also, as described above, someembodiments assume that all actual pins are on layer 2, while otherembodiments identify the actual layer of a pin that is stored. Inaddition, some embodiments define the layer of a virtual pin (i.e., apin that is set to account for a propagation of a route into alower-level child slot) to coincide with the layer of the propagatedpath for which the virtual pin was set.

[0433] After identifying the layers of the pins and the paths, theprocess 7000 then determines (at 7020) the difference between themaximum and minimum layers identified at 7015. This differencerepresents an estimate of the minimum number of vias that the routeneeds in the Current_Slot. If the Current_Slot is partitioned intosmaller slots, additional vias might be needed to define lower-levelroute(s) for the current route's net in and/or between the smallerslots. Some embodiments (1) might perform a statistical study to guessthe number of vias needed to define routes in slots with certain numberof pins and paths, and then (2) might use the results of thisstatistical study to obtain a better estimate of the number of viasneeded in the Current_Slot. At 7025, the process 7000 returns the numberof vias in the slot, and then ends.

[0434] Once the process 7000 returns the number of vias in the slot, theprocess 6900 marks (at 6910) the Current_Slot as one that has beenvisited. It then selects (at 6915) one of the current route's paths thatare incident on the Current_Slot. Next, at 6920, it identifies the otherslot that the path selected at 6915 is incident upon. At 6925, theprocess determines whether it has examined the slot identified at 6920(i.e., determines whether this slot is marked visited). If not, theprocess 6900 recursively calls itself at 6930. This recursive callspecifies the slot identified at 6920 as the Current_Slot for theprocess that is recursively initiated at 6930. At 6930, the process 6900increments the number of vias by the value that the recursively-calledprocess returns.

[0435] From 6930, the process 6900 transitions to 6935. This processalso transitions to 6935 when it determines (at 6925) that the otherslot identified at 6920 was previously visited. At 6935, the process6900 determines whether there are any other paths incident upon theCurrent_Slot. If so, the process 6900 returns to 6915 to select anotherincident path. If not, the process 6900 returns the computed number ofvias.

[0436] As indicated in equation (H) above, the objective function'sTotal_Via_Number not only depends on the number of vias for each tree,but also on a conversion factor that scales the number of vias toreflect their importance relative to the wirelength. This conversionfactor can be obtained by normalizing the wirelength and via costs tothe number of tracks, as indicated in equation (I) below.$\begin{matrix}{X = {50*{\frac{5}{N}.}}} & (I)\end{matrix}$

[0437] In this equation, X represents the conversion factor, 5represents the cost of a Manhattan path, N represents the number oftracks per Manhattan path at the current recursion level, and 50 is apenalty cost associated with using a via. This penalty is measured interms of the number of tracks that the router would prefer to detourrather than use a via. In some embodiments, a designer can modify thispenalty cost. Also, in some embodiments, each Manhattan path represents8 tracks at the Gcell level (i.e., N equals 8 at the Gcell level). Also,conversion factor X is different for each hierarchical routing levelsince N is different for each level.

[0438] b. Constraints

[0439] Different embodiments define different constraints. Theembodiments that use the LP formulation described below define the threeconstraints. First, for each net N, the LP solver must only choose 1tree from the selection set for the net N. This is expressed as aconstraint on the sum of the values of the tree variables for the net(i.e., the sum of these values must equal 1), as indicated in theequation below:

netN: xN _(—) A+xN _(—) B+ . . . xN _(—) Q=1.

[0440] The LP solver will assign a value between 0 and 1 to each routevariable. A “1” indicates an unambiguous selection of the tree, and a“0” indicates an unambiguous rejection of the tree. A value between 0and 1 means that the solver specified a set of selections that need tobe resolved by the randomized rounding. Some embodiments might notexpress the need to select only one tree for each net as a constraintbut rather as a relationship to generate candidate LP solutions for eachnet.

[0441] The second constraints relates to the congestion of the paths inthe current slot. In some embodiments, the minSlack has to be greaterthan a specified amount. As mentioned above, the minSlack is thesmallest tolerable slack across all the paths. The slack in each pathequals the capacity of the path minus wireflow and blockages across thepath. For the first iteration of the LP solver, some embodiments specifythat the minSlack has to be zero or greater. If the LP solver cannotsolve the formulated problem in its first iteration, some embodimentsreformulate the LP problem so that the minSlack is a large negativenumber, in order to remove the minimum slack as a constraint. However,in some embodiments, this reformulation makes the Min_Slack component ofthe objective function the primary component of this function, asdescribed above.

[0442] Third, the capacity of certain regions needs to be properlyshared among the paths that can traverse these regions. For instance,given the models illustrated in FIGS. 61, 62, and 63, diagonal andManhattan paths must properly share the capacity of overlapping diagonalregions. This is because when the solver is part of a global router thatproduces global routing results, its global-routing output needs to beconverted into boundary pin assignments that can be used by a detailedrouter.

[0443]FIGS. 71 and 72 illustrate the need for this sharing constraint atthe Gcell level. At the Gcell level, some embodiments define thecapacity of the Manhattan paths crossing between the Gcells to be 8tracks wide. FIG. 71 shows 8 such tracks across edge E11. Also, someembodiments assume that the pitch on the diagonal layers is the same asthe pitch on the Manhattan layers and therefore assume that the capacityof a diagonal path is simply square root of two times the capacity of aManhattan path. Under such assumptions, the capacity of a diagonal paththat crosses the above-described Gcells is 11 tracks. FIG. 71illustrates the 11-track-wide capacity of path P32.

[0444] Vertically or horizontally adjacent diagonal paths haveoverlapping routing regions. For instance, FIG. 72 illustrates twovertically adjacent diagonal paths P32 and P26 that share the capacity(i.e., the five diagonal tracks) of shared diagonal region 7205. Thefive diagonal tracks of diagonal region 7205 are also shared with theManhattan path P4 that crosses this region, because, according to themodel of FIG. 61, a Manhattan path can traverse between two slots notonly in the Manhattan direction but also in the diagonal directions.

[0445] To account for the shared capacity of the diagonal regions, theLP formulation below defines three sets of constraints, with each sethaving two constraints, one for +45° paths and one for −45° paths. Thesethree set of constraints are: (1) diagonal pair constraints, (2) mixedtriplet constraints, and (3) diagonal triplet constraints. In thediscussion below, +45° paths are referred to as East paths, while −45°paths are referred to as West paths.

[0446]FIG. 73 illustrates the first type of constraint, i.e., thediagonal pair constraints. This figure illustrates eight constraineddiagonal pairs that have been defined about paths P27 and P36. Two ofthese are 7305, which includes horizontally-adjacent east paths P36 andP38, and 7310, which includes vertically-adjacent west paths P27 andP33.

[0447] The diagonal pair constraints can be classified as interior orperiphery constraints depending on whether one of the diagonal paths ofthe pair is on the periphery of the current slot. Specifically, bothpairs 7305 and 7310 are interior diagonal pairs. An interior diagonalpair includes two diagonal paths that are horizontally or verticallyadjacent and that are either from the set of east paths P24, P26, P28,P30, P32, P34, P36, P38, and P40 or from the set of west paths P25, P27,P29, P31, P33, P35, P53, P39, P41. Three other interior constraineddiagonal pairs illustrated in FIG. 73 are (1) pair 7315, which includespaths P36 and P30, (2) pair 7320, which includes paths P27 and P25, and(3) pair 7325, which includes paths P27 and P29.

[0448]FIG. 73 also illustrates three constrained periphery diagonalpairs. A periphery diagonal pair includes two horizontally- orvertically-adjacent diagonal paths where (1) both are either east pathsor west paths, and (2) one of the paths is one of the interior pathsP24-P41 and the other path is on the periphery edge of the slot. Thethree constrained periphery diagonal pairs illustrated in FIG. 73 are:(1) pair 7330, which includes paths P36 and 7335, (2) pair 7340, whichincludes paths P36 and 7345, and (3) pair 7350, which includes paths P27and 7355.

[0449] The two paths in each constrained diagonal pair share severaltracks (i.e., several tracks represented by one path are the same asseveral tracks represented by the other path). Accordingly, someembodiments constrain the congestion about each constrained diagonalpair to account for this sharing.

[0450] For instance, some embodiments constrain the congestion about aninterior diagonal pairs to be 1.5 times the capacity of either path inthe pair, when the two paths in the pair share about half of the tracks.By way of example, when the two interior diagonal paths P36 and P38 areeach 11 tracks wide and share 5 tracks with each other, some embodimentsspecify a constraint that the congestion about these paths must at mosttake 16 tracks (i.e., specify that wireflow across P36 plus wireflowacross P38 plus any blockages at most take 16 tracks).

[0451] Some embodiments also constrain the congestion (i.e., the totalwireflow and blockages) about a periphery diagonal pair to be 1.5 timesthe capacity of the interior diagonal path, when the two paths in thepair share about half of the tracks. For example, in some embodiments,the paths in a periphery diagonal pair 7330 are each 11 tracks wide atthe Gcell level. At this level, these paths share 5 tracks with eachother. Accordingly, for the Gcell level, some embodiments specify aconstraint that the congestion about these path must be 16 tracks orless (ie., specify that wireflow across P36 plus wireflow across 7335plus any blockages at most take 16 tracks).

[0452] Unlike an interior diagonal pair constraint that requires the LPsolver to compute the congestion about both the diagonal paths in thepair, a periphery diagonal pair constraint only requires the LP solverto compute the congestion about the interior diagonal path of theperiphery pair for an LP solution. This is because the congestion aboutthe periphery path of a periphery diagonal pair is computed during thepropagation operation that preceded the current solve operation.

[0453] For instance, the congestion about the periphery path 7335 forthe periphery diagonal pair 7330 might have been computed when theManhattan or diagonal path incident upon slot 7360 or the diagonal pathincident upon a slot neighboring slot 7360 were propagated down into thechild slot 7365 of slot 7360. Some embodiments keep a record of thecapacity of a propagated path between a child slot of a first parentslot and the child slot of a second parent slot that is adjacent to thefirst parent slot. Some embodiments maintain such a record by (1)creating slot pair record for adjacent child slots of adjacent parentslots, (2) storing the identity of the two adjacent child slots in theslot-pair record, (3) initializing a capacity field that represents thecapacity of the propagated periphery path between the two child slots,and (4) decrementing this capacity for use and blockages. Theseembodiments then identify the capacity of each periphery path byretrieving the slot-pair record that stores this capacity. Someembodiments retrieve the slot-pair record by using the identity of thetwo child slots on which the periphery path is incident (i e., thecurrent slot's child slot, and the neighboring slot's child slot).

[0454]FIG. 74 illustrates the second type of constraint, i. e., themixed triplet constraints. This constraint is similar to the first typeof constraint, the diagonal pair constraint, except that the mixedtriplet constraint constrains the congestion about an adjacent co-lineardiagonal path pair plus the Manhattan path between the diagonal pair.

[0455]FIG. 74 illustrates eight constrained mixed triplets, fourinvolving path P36 and four involving path P27. Like the diagonal pairconstraints, the mixed triplet constraints can be classified as interioror periphery constraints depending on whether one of the diagonal pathsof the triplet is on the periphery of the current slot.

[0456] The four constrained mixed triplets about path P36 in FIG. 74are: (1) interior mixed triplet 7405, which includes paths P21, P36 andP38, (2) periphery triplet 7410, which includes paths P9, P36 and 7445,(3) periphery triplet 7415, which includes paths P20, P36 and 7455, and(4) interior mixed triplet 7420, which includes paths P6, P36 and P30.

[0457] The four constrained mixed triplets about path P27 in FIG. 74are: (1) interior mixed triplet 7425, which includes paths P14, P27 andP29, (2) interior triplet 7430, which includes paths P4, P27 and 7433,(3) interior triplet 7435, which includes paths P13, P27 and P25, and(4) interior mixed triplet 7440, which includes paths P1, P27 and 7450.

[0458] The three paths in each constrained mixed triplet share severaltracks. For instance, given the diagonal wire model representation ofFIGS. 62 and 63, several tracks represented by one of the diagonal pathsare the same as several tracks represented by the other diagonal path.Also, given the Manhattan wire-model representation of FIG. 61, aManhattan path can be propagated into lower-level child slots throughseveral diagonal paths that compete for the same tracks as the diagonalpaths neighboring the Manhattan path. Accordingly, some embodimentsconstrain the congestion about each constrained mixed triplet to accountfor this sharing.

[0459] For instance, when the two diagonal paths in the pair share abouthalf of the tracks, some embodiments constrain the congestion about aninterior mixed triplet to be 1.5 times the capacity of one of thediagonal path's in triplet plus the capacity of the Manhattan path inthe Manhattan direction only. For example, at the Gcell level, the twointerior diagonal paths P36 and P38 are each 11 tracks wide and share 5tracks with each other, while the Manhattan path P21 is 8-tracks wide inthe Manhattan direction and 5tracks wide in the East direction (i.e.,path P21 can use 8 tracks on the vertical layer of wiring and 5 tracksin the East layer wiring). Accordingly, at the Gcell level, some ofthese embodiments specify a triplet constraint that the congestion(i.e., wireflow plus blockages) about paths P21, P36, and P38 must atmost take 24 tracks (i.e., the 16 available East tracks plus the 8available vertical tracks).

[0460] Other embodiments might constrain the congestion an interiormixed triplet to be 1.5 times the capacity of one of the diagonal path'sin triplet plus the capacity of the Manhattan path in the Manhattandirection and the opposite diagonal direction, when the two diagonalpaths in the pair share about half of the tracks. So, for the aboveexample (where, at the leaf-slot level, the two interior diagonal pathsP36 and P38 are each 11 tracks wide and share 5 tracks with each other,while the Manhattan path P21 is 8-tracks wide in the Manhattandirection, 5-tracks wide in the East direction, and 5-tracks wide on theWest direction), some of these embodiments specify a triplet constraintthat the congestion about paths P21, P36, and P38 must at most take 29tracks (i.e., the 16 available East tracks plus the 8 available verticaltracks plus 5 available West tracks).

[0461] In addition, like the periphery diagonal pair constraints, theperiphery mixed triplet constraints are analyzed like the interiorconstraints in some embodiments, except for the computation of thecongestion about the periphery path of the triplet during the priorpropagation operation. In other words, an interior mixed tripletconstraint requires the LP solver to compute the congestion about boththe triplets diagonal paths and Manhattan path for an LP solution. Theperiphery mixed triplet constraint only requires the LP solver tocompute the congestion about the triplet's interior diagonal path andthe Manhattan path for an LP solution. The LP solver can retrieve thecongestion about the periphery path from the slot-pair record for thetwo child slots that the periphery path traverses.

[0462]FIG. 75 illustrates the third type of constraint, i.e., thediagonal triplet constraints. This constraint is similar to the firsttype of constraint, the diagonal pair constraint, except that thediagonal triplet constraint constrains the congestion about threeco-linear diagonal paths instead of two. FIG. 75 illustrates fourconstrained diagonal triplets, two involving path P36 and two involvingpath P27.

[0463] Like the diagonal pair constraints, the diagonal tripletconstraints can be classified as interior or periphery constraintsdepending on whether one of the diagonal paths of the triplet is on theperiphery of the current slot. The four constrained diagonal triplets inFIG. 75 are: (1) periphery diagonal triplet 7505, which includes pathsP36, P38 and 7530, (2) periphery diagonal triplet 7510, which includespaths P30, P36 and 7525, (3) interior diagonal triplet 7515, whichincludes paths P25, P27 and P29, and (4) periphery diagonal triplet7520, which includes paths P27, P33, and 7535.

[0464] Given the diagonal wire model representation of FIGS. 62 and 63,the middle path in the triplet shares several tracks with the other twodiagonal paths. Accordingly, some embodiments constrain the congestionabout each constrained diagonal triplet to account for this sharing.

[0465] When the middle diagonal path in the triplet shares about half ofits tracks with one of the other diagonal paths and shares the otherhalf with the other diagonal path, some embodiments constrain thecongestion about a diagonal triplet to be 2.0 times the capacity of oneof the diagonal path's in triplet. For example, in some embodiments atthe Gcell level, each diagonal path is 11 tracks wide, and shares 5tracks with each neighboring co-linear diagonal path. Accordingly, forthe Gcell level, some embodiments specify a triplet constraint that thecongestion (i.e., wireflow plus blockages) about the three diagonalpaths in a triplet (e.g., about paths P25, P27, and P29) must at mosttake 22 tracks.

[0466] In addition, like the periphery diagonal pair and mixed tripletconstraints, the periphery diagonal triplet constraints are analyzedlike the interior constraints in some embodiments, except for thecomputation of the congestion about the periphery path of the tripletduring the prior propagation operation. Specifically, an interiordiagonal triplet constraint requires the LP solver to compute thecongestion about all the diagonal paths of the triplet. On the otherhand, a periphery diagonal triplet constraint only requires the LPsolver to compute the congestion about the triplet's interior diagonalpaths. The LP solver can retrieve the congestion about the peripherydiagonal path from the slot-pair record for the two child slots that theperiphery path traverses.

[0467] One of ordinary skill will realize that other embodiments mightdefine other constraints. For instance, some embodiments might define amixed quintuplet constraint, which constrains the congestion about aManhattan path and the two pairs of adjacent co-linear diagonal paths.One such quintuplet would include vertical path P13, −45° diagonal pathsP25 and P27, and +450 diagonal paths P24 and P26.

[0468] The Manhattan path in each quintuplet would share several trackswith the quintuplet's diagonal paths. In addition, the paths in eachparallel diagonal pair share several tracks. Accordingly, when the twodiagonal paths in each pair share about half of the tracks, someembodiments constrain the congestion about a mixed quintuplet to beabout 3.0 times the capacity of one of the diagonal path's in quintupletplus the capacity of the Manhattan path in the Manhattan direction only.For example, in some embodiments at the leaf-slot level, each diagonalpath is 11 tracks wide and shares 5 tracks with its adjacent co-lineardiagonal path, while each Manhattan path is 8-tracks wide in theManhattan direction. Accordingly, at the Gcell level, some of theseembodiments specify a quintuplet constraint that the congestion aboutthe quintuplet must at most take 40 or 41 tracks, depending on whetherthe capacity about each parallel diagonal pair is truncated or not.

[0469] C. Formulation

[0470] In some embodiment, the solver 3930 formulates the LP problembased on the above-described objective function and constraints. Theformulation of the LP problem in some of these embodiments is asfollows:

[0471] [prepSolverILP (slot)]

[0472] initialize variable minSlack to 0 // paths are not allowed to beover-constrained

[0473] initialize variable lenAndViaWeight to 100 // initially lengthand via count has priority over minimizing slack

[0474] initialize variable minSlackWeight to −1

[0475] while (! done)

[0476] declare & initialize the LP solver

[0477] declare objective row, and name it “objective”

[0478] for each identified set of trees for a net in the variable thatstores all sets of identified trees for all nets in the current slot

[0479] if set is not empty (i.e., the set contains tree selection setfor a net in this slot)

[0480] retrieve net index, N, from first tree record in set

[0481] declare a constraint, “rN”, to constrain the solver to chooseonly 1 tree for this net

[0482] for each path, N, in the slot

[0483] declare a constraint, “usageN”, to define a variable summing thetotal usage of this path

[0484] declare a constraint, “eSlackN”, to define a maximum slack valueover all paths

[0485] declare a constraint, “mxuseN”, to define a maximum usage valueover all paths

[0486] if path N is a Manhattan path

[0487] declare a constraint, eMtplN, to constrain the sum of usages ofManhattan path N, and adjacent pair “east” paths // this constraint isfor the mixed triplet that includes path N and 2 diagonal east pathsadjacent to it

[0488] declare a constraint, wMtplN, to constrain the sum of usages ofManhattan path N, and adjacent “west” paths // this constraint is formixed triplet that includes path N and 2 diagonal west paths adjacent toit

[0489] declare a constraint, epairN, to constrain the sum of usages ofthe 2 “east” paths adjacent to path N // this constraint is for diagonalpair that includes 2 diagonal east paths adjacent to path N

[0490] declare a constraint, wpairN, to constrain the sum of usages ofthe 2 “west” paths adjacent to path N // this constraint is for diagonalpair that includes 2 diagonal west paths adjacent to path N

[0491] declare a constraint, eDtplN, to constrain the sum of usages ofthe 2 “east” paths adjacent to path N, plus a 3rd east path below or tothe left of the bottom-most/left-most adjacent east path // thisconstraint is for diagonal triplet that includes 2 diagonal east pathsadjacent to path N plus a 3rd east path below or to the left of thebottom-most/left-most adjacent east path

[0492] declare a constraint, wDtplN, to constrain the sum of usages ofthe 2 “west” paths adjacent to path N, plus a 3rd west path below or tothe left of the bottom-most/left-most adjacent west path HI thisconstraint is for diagonal triplet constraints that includes 2 diagonalwest paths adjacent to path N plus a 3rd west path below or to the leftof the bottom-most/left-most adjacent west path

[0493] define a constraint, “Min_Slack”, to limit the value of theminimum slack across all paths

[0494] define a constraint, “tLen”, to define a variable summing thelength costs of all chosen trees

[0495] define a constraint, “tVias”, to define a variable summing thevia costs of all paths

[0496] // at this point, all the “rows” of the LP are declared. Now wecontinue by filling in the columns (i.e. declaring the variables)

[0497] for each set of trees, treeset, for a net in the variable,m_sols, that stores the sets of trees for all nets in the current slot

[0498] identify the net to which this set of trees belongs

[0499] for each tree in treeset

[0500] create a variable, “xN_T”, where N is the net index and T is theordinal number of the tree, to represent this tree

[0501] declare xN_T to be present with factor 1.0 in constraint “rowN”

[0502] identify wirelength cost of tree xN_T // computed by process 6400described above

[0503] compute the number of vias, “nVias”, required to embed this tree// uses the processes 6800-7000 described above

[0504] declare xN_T to be present with factor estLen in constraint“tLen”

[0505] declare xN_T to be present with factor nVias times X inconstraint “tVias” // where X is the conversion factor that wasdescribed above by reference to equation (I)

[0506] for each path, E, in the slot

[0507] if this tree uses path E

[0508]  declare xN_T to be present with factor −1.0 in constraint usageE

[0509] for each path, E, in this slot

[0510] create variable, “uE”, where E is an integer identifier of thepath

[0511] declare uE to be present with factor 1.0 in constraint “usageE”

[0512] declare uE to be present with factor −1.0 in constraint “mxuseE”

[0513] declare uE to be present with factor 1.0 in constraint “eslack”

[0514] if path E is a Manhattan path

[0515] declare uE to be present with factor 1.0 in constraint “eMtplE”

[0516] declare uE to be present with factor 1.0 in constraint “wMtplE”

[0517] retrieve two pairs of diagonal paths adjacent to path E

[0518] for each path, A, in the pairs

[0519] if the direction of path A is “east”

[0520]  declare uA to be present with factor 1.0 in constraint eMtplE

[0521]  declare uA to be present with factor 1.0 in constraint epairE

[0522] if the direction of path A is “west”

[0523]  declare uA to be present with factor 1.0 in constraint wMtplE

[0524]  declare uA to be present with factor 1.0 in constraint wpairE

[0525] retrieve triple of diagonal paths adjacent to path E

[0526] for each path, B, in the triple

[0527] if the direction of path B is “east”

[0528]  declare uB to be present with factor 1.0 in constraint eDtplE

[0529] if the direction of path B is “west”

[0530]  declare uB to be present with factor 1.0 in constraint wDtplE

[0531] create variable, “slack”

[0532] declare slack to be present with factor 1.0 in constraint“Min_Slack”

[0533] declare slack to be present with factor “minSlackWeight” in theobjective function

[0534] for each path, E, in this slot

[0535] declare slack to be present in constraint with factor 1.0 inconstraint “eslackE”

[0536] if path E is a manhattan path

[0537] declare slack to be present with factor 1.0 in constraint“epairE”

[0538] declare slack to be present with factor 1.0 in constraint“wpairE”

[0539] declare slack to be present with factor 1.0 in constraint“eMtplE”

[0540] declare slack to be present with factor 1.0 in constraint“wMtplE”

[0541] declare slack to be present with factor 1.0 in constraint“eDtplE”

[0542] declare slack to be present with factor 1.0 in constraint“wDtplE”

[0543] create variable, “tV”

[0544] declare tV to be present with factor 1.0 in constraint “tVias”

[0545] declare tc to be present with factor “lenAndViaWeight” in theobjective function

[0546] create variable “tL”

[0547] declare tl to be present with factor 1.0 in constraint “tLen”

[0548] declare tl to be present with factor “lenAndViaWeight”in theobjective function

[0549] // up to here, we have declared constraints and filled in theleft hand sides of their equations. Now, we'll set their right handsides

[0550] for each set of trees, treeset, for a net in the variable,m_sols, that stores the sets of trees for all nets in the current slot

[0551] if set is not empty (set contains tree selection set for a net inthis slot)

[0552] retrieve net index, N, from first tree record in set

[0553] set RHS of constraint, “rN”, to equal to 1.0

[0554] for each path, E, in this slot

[0555] set th e rhs of constraint “usageE” equal to 0.0

[0556] set the rhs of constraint “mxuseE” equal to 0.0

[0557] retrieve capacity estimate, cap(E), produced by subtractingestimated path use (computed by process 5500) from estimate unblockedvalue (computed by process 5400)

[0558] set the rhs of constraint “eslackE” to cap(E)

[0559] if path E is a manhattan path

[0560] calculate capacity estimate for sharing constraint eMtplE,capeMtplE, and set the rhs of constraint eMtplE to this capacity

[0561] calculate capacity estimate for sharing constraint wMtplE,capwMtplE, and set the rhs of constraint wMtplE to this capacity

[0562] calculate capacity estimate for sharing constraint epairE,capepairE, and set the rhs of constraint epairE to this capacity

[0563] calculate capacity estimate for sharing constraint wpairE,capwpaire, and set the rhs of constraint wpairE to this capacity

[0564] calculate capacity estimate for sharing constraint eDtplE,capeDtplE, and set the rhs of constraint eDtplE to this capacity

[0565] calculate capacity estimate for sharing constraint wDtplE,capwDtplE, and set the rhs of constraint wDtplE to this capacity

[0566] set the rhs of constraint “tVias” equal to 0.0

[0567] set the rhs of constraint “tLen” equal to 0.0

[0568] set the rhs of constraint “Min Slack” equal to variable minSlack

[0569] solve the LP

[0570] if no solution exists // remove hard constraint on minimum slack,reset weights such that slack is priority over length and via count

[0571] set variable minSlackWeight=−500

[0572] set variable minSlack=−1000

[0573] set variable lenAndViaWeight=1

[0574] if solution was found

[0575] exit while loop

[0576] As indicated above, the third-to-last statement of theformulation tells (at 5325) the LP solver to solve the problem. The LPsolver then tries to solve this problem. If the LP solver fails to solvethis problem in the first iteration through the while loop describedabove, the formulation above changes values of certain constants so thatthe minimum slack is no longer much of a constraint but rather serves asthe primary component of the objective function. Specifically, thechange to the constants minSlackWeight, minSlack, and lenAndViaWeighteffectively makes the capacity constraint (which is the only constraintthat would cause the first attempt to fail) ineffective. The LP solverthen tries to solve the problem again. The change to the constantsminSlackWeight, minSlack, and lenAndViaWeight ensures that the secondattempt will produce a solution. One of ordinary skill will understandthat other embodiments might change the value of these variables moreincrementally to find solutions with different characteristics. However,such incremental changes might reduce the speed of the solver.

[0577] The solution that the LP solver returns is one that meets all theconstraints and produces the lowest objective-function output. Thereturned solution may include real numbers for each tree variable xN_C .For instance, if the solver submitted three routes to the LP solver, theLP solver might return a score of 0.8 for one route, 0.1 for the otherroute, and 0.1 for the last route.

[0578] As mentioned above, the process 5300 converts (at 5330) this LPsolution into an ILP solution, i.e., a solution that specifies a 0 or 1as the value of each tree variable xN_C. Also, as mentioned above, someembodiments use randomized rounding to perform this conversion. Based onthe set of routes selected at 5330, the solver 3930 stores (at 5335) a42-bit selected route string in each slot-net data structure of thecurrent slot. This 42-bit string specifies the paths in the current slotthat the selected route of the net takes.

[0579] One of ordinary skill will understand that despite the abovedescription of the solver, other embodiments might use differentapproaches to solve the routing problem at any particular level of therouting hierarchy. For instance, some embodiments might define objectivefunctions and constraints in a different manner than those describedabove. For instance, some embodiments might use the cost of a route as aconstraint, and have the objective function simply minimize congestion.Also, instead of using an LP solver to generate an LP solution andconverting the LP solution into an ILP solution, other embodiments usean ILP solver to generate an ILP solution. Yet other embodiments use asequential approach to embed routes for each net in the current slot.

[0580] F. Propagator

[0581] After the solver specifies the route for each slot-net of thecurrent slot, the slot manager 3925 calls the propagator 3935 when thecurrent slot is not at a leaf slot. The propagator then determines howthe routing paths specified by the solver for the current routing levelpropagate down into the child slots of the current slot. For slots thatare after the top-level slot but before the leaf-level slot, thepropagator also performs a follow-up propagation operation thatpropagates the paths specified by the propagator at the previous routinglevel one level further down. For each net in the current slot, thepropagator might have to modify the net's pin distribution within eachchild slot to account for the propagations that it identifies.

[0582] Two different propagators are described below. The firstpropagator enumerates several propagation solutions for each net's routeand then uses the LP solver 3945 and ILP converter 3950 to select apropagation solution for each net. The second propagator, on the otherhand, is a sequential propagator that uses a greedy approach to selectand embed a propagation for the route of each net in the current slot.In the embodiments described below, both these propagators also use asequential propagator to perform the follow-up propagation, whenapplicable.

[0583] Some embodiments use the first propagator when they use theseven-permutation propagation model of FIG. 62 for diagonal paths, anduse the second propagator when they use the nineteen-permutationpropagation model of FIG. 63 for diagonal paths. Some of theseembodiments use the ten-permutation propagation model of FIG. 61 forManhattan paths, in conjunction with either of these models.

[0584]1. ILP Propagator

[0585] Like the solver, the ILP propagator enumerates and costs severalpropagation solutions for each net's route into the affected childslots. The propagator then formulates an LP problem and feeds thesesolutions to the LP solver 3945, which, in turn, returns a number ofreal-number solutions. These real-number solutions are then convertedinto integer solutions by the ILP solver 3950. These integer solutionsspecify a particular configuration for each net within each affectedchild slot, and the propagator stores each net's configuration in thenet's slot-net data structure for the affected child slot.

[0586]FIG. 76 illustrates a process 7600 that the ILP propagatorperforms in some embodiments. In some embodiments, this process startswhen the slot manager calls the propagator and supplies it with acurrent slot. The process 7600 initially estimates (at 7605) theavailability of each propagation possibility of each path. One manner ofestimating the availability of the propagations will be described belowby reference to FIGS. 77 and 78.

[0587] After estimating the availability of each propagation possibilityof each path, the process 7600 enumerates and costs (at 7610) allpropagation permutations for each slot-net in the current slot. Onemanner of enumerating and costing the propagations will be describedbelow by reference to FIGS. 79 and 80.

[0588] After enumerating and costing the potential propagationpermutations, the LP propagator formulates an LP problem for the LPsolver 3945. One manner of formulating the LP propagation problem willbe described below in Section VI.F.1.d. The process 7600 then converts(at 7625) the LP solution returned by the LP solver to an ILP solution.In some embodiments, the process performs randomized rounding to makethis conversion. One manner of performing randomized rounding wasdescribed above in Section VI.E.

[0589] Based on the propagations specified at 7625, the process thenmodifies (at 7630) the 16-bit pin distributions of the slot-nets in thechild slots of the current slot, when necessary. If at this stage thereis no slot-net data structure to modify for a particular net, thepropagator will instantiate one and record the 16-bit pin distributionin it.

[0590] When the current slot's level is at least two levels above theleaf level, the process 7600 adds (at 7635) the propagation paths thatit identified at 7625 to the follow-up propagation list for the nextlower recursion level. When the current slot's level is after the toplevel but before the leaf level, the propagator then performs (at 7640)a follow-up propagation operation. This operation propagates the routingpaths specified by the propagator at the previous routing level onelevel further down. One manner of performing follow-up propagation isexplained below by reference to FIGS. 65 and 81.

[0591] When the current slot's level is the level immediately before theleaf level (i.e., when the current slot is a grandparent of Gcells), theprocess 7600 next calls (at 7645) the saver to link to the dBNets 4110the path data structures of the propagation paths specified at 7625 and,when applicable, for the propagation paths specified at 7640. Theprocess then ends.

[0592] a. Estimating Congestion of the Propagations.

[0593] As mentioned above, the process 7600 estimates (at 7605) theremaining availability of each propagation possibility for each path inthe current slot. In some embodiments, the process 7600 computes thisestimate by (1) estimating the blocked capacity of each propagation ofeach path, (2) estimating the use of each propagation of each path, and(3) summing each propagation's blocked capacity and use. The estimationof the blocked capacity of each propagation is described below byreference to FIG. 77, while the estimation of the use of eachpropagation is described below by reference to FIG. 78.

[0594] (1) Estimated Blocked Capacity of Each Path.

[0595]FIG. 77 illustrates a process 7700 for estimating the blockedcapacity of each propagation of each path in the current slot. Thepropagation process 7600 performs process 7700 at 7605. Initially, thisprocess allocates (at 7702) a data structure (e.g., a matrix) that hasat least one field for storing the blocked capacity of each propagationof each path in the current slot. At 7702, the process also initializeseach field in the data structure to 0.

[0596] At 7704, the process selects a circuit module in the currentslot's list of circuit modules. The process then retrieves (at 7706) thecircuit macro for the selected circuit module. It then selects (at 7708)an obstacle on the circuit macro, and computes (at 7710) the boundingbox of the selected obstacle.

[0597] Next, the process (at 7712) selects one of the 42 paths of thecurrent slot. It then selects (at 7714) one of the propagations of thepath selected at 7712. The process next determines (at 7716) whetherpath selected at 7712 is on the same layer as the obstacle selected at7708.

[0598] If the selected path's layer matches the selected obstacle'slayer, the process calculates (at 7726) the bounding box of the selectedpropagation. At 7726, the process also calculates the area of thebounding box of the propagation. The process next identifies (at 7728)the intersection of the selected propagation's bounding box and theselected circuit module's bounding box, and calculates (at 7730) thearea of this intersection. The process computes (at 7732) an obstructionfactor by dividing they calculated intersection area by they calculatedpropagation area. The process next multiplies (at 7734) the obstructionfactor by the default propagation capacity, and then adds (at 7736) theresult of this multiplication to the propagation's blocked capacity thatis stored in the data structure allocated at 7702. The process thentransitions to 7718, which is described below.

[0599] If the process determined at 7716 that the selected path's layeris not the same as the selected obstacle's layer, the processtransitions to 7718. At 7718, the process determines whether it hasexamined all the propagations for the path selected at 7712. If not, theprocess returns to 7714 to select another propagation for the selectedpath. On the other hand, if the process determines (at 7718) that it hasexamined all the propagations for the path selected at 7712, the processdetermines (at 7720) whether it has examined all the paths of thecurrent slot. If not, the process returns to 7712 to select another pathof the current slot.

[0600] Alternatively, if the selected path is the last path of thecurrent slot, the process determines (at 7722) whether it has examinedall the obstacles of the circuit module selected at 7704. If not, theprocess transitions to 7708 to select another obstacle of the selectedcircuit module. Otherwise, the process determines (at 7724) whether ithas examined all the circuit modules in the current slot. If not, theprocess transitions to 7704 to select another circuit module in thecurrent slot. However, if the process examined all the circuit modulesin the current slot, the process ends.

[0601] b. Estimated Use of Each Path Propagation.

[0602]FIG. 78 illustrates a process 7800 for estimating the use of eachpropagation of each path in the current slot. This process starts eachtime the propagator calls it at 7605. In some embodiments, the process7800 receives from the propagator a data structure (e.g., a matrix) offloating-point variables for storing the estimated use of eachpropagation. In other embodiments, the process 7800 does not receivesuch a data structure, but rather creates this structure when it starts.In some embodiments, the received or created data structure has at leastone entry for each propagation possibility.

[0603] As shown in FIG. 78, the process 7800 initially selects (at 7805)one of the child slots of the current slot. It then calls (at 7810) thepath-use estimating process 5500 of FIG. 55 for the selected child slot.The path-use estimating process 5500 computes and returns an estimatedusage value for each path of the selected child slot. As the estimatingprocess 5500 was described above, it will not be described here in orderto not obscure the description of the invention with unnecessary detail.

[0604] After 7810, the process 7800 determines (at 7815) whether it hascomputed the path usage values for all the child slots of the currentslot. If it has not, it returns to 7805 to select another child slot,and computes (at 7810) the path-usage values for the newly-selectedchild slot. When the process determines (at 7815) that it has examinedall the current slot's child slots, it selects (at 7820) one of the 42paths in the current slot.

[0605] At 7825, the process selects one of the propagations for theselected path. It then computes (at 7830) an estimate of the use of theselected path propagation based on that path-usage values of theneighboring child slot paths. For instance, in some embodiments of theinvention, the process uses the formula below to compute the usage ofpropagate 0 of path 1:prop_0-path_1_use = (1/2) * (1/2 * path[1][2] + 1/3 * path[1][1] + 1/6 * path[1][0] + 1/2 * path[2][0] + 1/3 * path[2][1] + 1/6 * path[2][2]),

[0606] where path[i][j] refers to the usage of path j of child slot i.Similar equations can be used to analogously define the propagationusage values for the other propagation possibilities.

[0607] The equation above defines a propagation usage value forpropagation 0 of path 1 between child slots 1 and 2 in terms of thecongestion within child slots 1 and 2. This equation only examines thehorizontal path in the child slots that are in line with propagation 0of path 1. Specifically, it examines the component usage value ofpropagate 0 of path 1 in terms of the horizontal paths 0, 1, and 2 ofchild slots 1 and 2. The summation of the usage values in both childslots 1 and 2 is multiplied by ½ to reflect that the capacity ofpropagation 0 of path 1 in the current slot is equally influenced by thecapacities of the child paths in child slots 1 and 2.

[0608] The multipliers ½'s, ⅓'s and ⅙'s are used in the summation forboth child slots 1 and 2 for the following reasons. The objective is toguess how many wires can be pushed through a propagate path. Some ofthese wires will terminate immediately after crossing the propagatepath, while some will cross the entire width of the slot incident to thepath. It is assumed that there will be a uniform distribution of“endpoints” of the wires using the path, such that for propagate 0 ofpath 1 ¼ will terminate in slot 0 of child slot 2, ¼ will terminate inslot 1 of child slot 2, ¼ in slot 2 of child slot 2 and ¼ in slot 3 ofchild slot 2 and beyond. This means that ¾ of the wires that use thepath 1 will also use path 0 of child slot 2, {fraction (2/4)} will usepath 1 in child slot 2, and ¼ will use path 3 in child slot 2—whichgives a ratio of 3:2:1 (or {fraction (3/6)}, {fraction (2/6)}, ⅙) ofrelative impact of the usages of these 3 paths on the estimated use ofthe propagated path.

[0609] At 7835, the process determines whether it has examined all thepotential propagations of the path selected at 7820. If not, the processtransitions back to 7825 to select another propagation for the selectedpath, and computes (at 7830) an estimate of the use of thenewly-selected propagation.

[0610] When the process determines (at 7835) that it has examined allthe propagations for the path selected at 7820, the process determines(at 7840) whether it has examined all the paths of the current slot. Ifit has not examined all paths, the process transitions back to 7820 toselect another path of the current slot, and then performs operations7825-7835 to compute the use of the propagation possibilities of thenewly-selected path. When the process determines (at 7840) that it hasexamined all the paths in the current slot, the process ends.

[0611] c. Enumerating and Assigning costs for each propagation

[0612] After estimating the availability of each propagation possibilityof each path, the process 7600 enumerates and costs (at 7610) allpropagation permutations for each slot-net in the current slot. FIG. 79illustrates one manner of enumerating and costing the propagations.

[0613] As shown in FIG. 79, the process 7900 starts by selecting (at7905) a slot-net of the current slot. The process then initializes (at7910) 16 empty lists, one for storing the paths incident on a particularchild slot. The process next retrieves (at 7915) the route for theslot-net selected at 7905.

[0614] At 7920, the process selects one of the paths of the retrievedroute. It then identifies the two child slots corresponding to the endpoints of the selected path. The process adds the selected path to thepath list of each child slot identified at 7925. At 7935, the processdetermines whether it has examined all the paths of the route retrievedat 7915. If not, the process returns to 7920 to select another path ofthe route.

[0615] When the process determines that it has added all the paths ofthe route to their corresponding child slots' lists, the process selects(at 7940) one of the child slots of the current slot and retrieves thelist of paths of the selected child slot. The process selects the childslot at 7940 in order to enumerate and cost all the possible propagationpermutations of the selected slot-net in the selected child slot. At7945, the process retrieves the selected slot-net's pin distribution inthe selected child slot. At 7950, the process initializes an empty listto store all possible path propagation configurations in the child slotselected at 7940. At 7955, the process determines whether the selectedchild slot's path list is empty (i.e., whether the slot-net's route hasany paths that traverse the child slot). When the slot-net's route doesnot traverse the selected child slot, the process does not need toidentify propagation configurations for the slot-net's route through theselected child slot. Accordingly, the process transitions to 7985 todetermine whether it has examined all the child slots of the currentslot. The flow of the process 7900 from 7985 will be described below.

[0616] If the process determines (at 7955) that the slot-net's routetraverses the selected child slot and that it therefore needs toidentify propagation configurations for the slot-net's route in theselected child slot, the process 7900 performs 7960-7980 to enumerate,cost, and store all the possible propagation permutations of theselected slot-net in the selected child slot.

[0617] In some embodiments, the process 7900 uses a recursive functionto perform 7955-7980. This function identifies each path-propagationpermutation by (1) selecting one possible propagation for a path on theselected child slot's path list, (2) setting a virtual pin to accountfor the selected propagation, (3) recursively repeating the first twooperations for each of the subsequent paths on the path list when suchpaths exist. For each identified propagation permutation, the process7900 then performs 7970-7975 to cost and save each permutation, and addeach permutation to a list of propagation configurations.

[0618] More specifically, at 7960, the process 7900 identifies onepermutation of path propagations in the selected child slot. When theslot-net's route has only one path that is incident on the selectedchild slot, the identified propagation permutation is one of thepropagation possibilities for the path incident on the selected childslot. On the other hand, when the slot-net's route has more than onepath incident on the selected child slot, each identified permutation isa unique combination of propagations for each of the paths incident onthe selected child slot.

[0619] As illustrated in FIG. 61, a horizontal vertical path has tenpropagation possibilities in some embodiment of the invention. On theother hand, a diagonal path has seven propagation possibilities in someembodiment as illustrated in FIG. 62, while it has nineteen propagationpossibilities in other embodiments as illustrated in FIG. 63. One ofordinary skill will understand that other embodiments use otherpropagation models for horizontal, vertical, or diagonal paths.

[0620] After identifying one permutation of path propagations in theselected child slot, the process identifies (at 7965) a pinconfiguration that accounts for the path propagations of the permutationidentified at 7960. Such a pin configuration is the same as theslot-net's pin distribution in the selected child slot except that itmight include one or more virtual pins to account for path propagationsof the identified permutation.

[0621] The process then computes (at 7970) the cost of the pinconfiguration identified at 7965. In some embodiments, this cost is thewirelength cost of the route necessary for connecting the selected childslot's pins that are specified by the identified pin configuration. Asbefore, some embodiments retrieve this cost from a pre-tabulated tablethat specifies the cost of the optimal Steiner routes for each pinconfiguration, while other embodiments compute this cost in real timebased on the costs of the route paths.

[0622] At 7970, the process stores the identified propagationpermutation (i.e., the identified path propagations)_and its cost in aconfiguration record. The data structure for such a record isillustrated in FIG. 80. The propagator creates a list of this datastructure and uses this list to keep track of all the configurationsgenerated by the propagator. This data structure includes a reference tothe net's dbNet data structure. It also contains a child-slot identifierthat identifies for the propagator the identity of the child slot forthe configuration. This structure also includes a name from which pathpropagations can be derived. It further stores the wirelength cost and alist of paths.

[0623] After 7970, the process adds (at 7975) the configuration recordcreated at 7970 to a list of configuration for the selected child slot.The process then determines (at 7980) whether it has examined all thepath-propagation permutations in the selected child slot. As mentionedabove, some embodiments perform this determination as part of arecursive function that identifies all the path-propagationpermutations.

[0624] If the process determines (at 7980) that it has not examined allpath-propagation permutations, it identifies (at 7960) anotherpermutation and then costs and stores (at 7965-7975) this permutation.When the process has examined all path-propagation permutation, itdetermines (at 7985) whether it has examined all the child slots. Ifnot, the process returns to 7940 to select another child slot.

[0625] When the process determines (at 7985) that it has examined allthe child slots, the process determines (at 7990) whether it hasgenerated the propagation permutations for all slot-nets in the currentslot. If not, the process returns to 7905 to select another slot-net,and then performs subsequent operations to enumerate and cost thepropagation permutations for the newly-selected slot-net. The processends when it has examined all the slot-nets in the current slot.

[0626] d. LP Problem Formulation and Solving

[0627] The ILP propagator 3935 formulates the LP problem by providingthe LP solver 3945 with one or more objective functions, a number ofsolutions, and several constraints. The LP solver then needs to use theobjective functions to select the optimal solution in view ofconstraints.

[0628] The basic variables in the LP-propagation formulation are theconfiguration records, nXtYeApB . . . , where the lower case letters arekeywords (n=net; t=child slot; e=path; p=propagation), and theupper-case letters represent numbers (from 0 to the number of nets inthe design for ‘n’; from 0-15 for ‘t’; from 0-41 for ‘e’; and from 0-9for ‘p’).

[0629] This LP solver returns an LP solution that includes a real numbervalue for each configuration variable. As mentioned above, the process7600 then converts this LP solution into an ILP solution, i.e., asolution that specifies a 0 or 1 as the value of each configurationvariable. Instead of using an LP solver to generate an LP solution andconverting the LP solution into an ILP solution, other embodiments usean ILP solver to generate an ILP solution.

[0630] As mentioned above, some embodiments use as the LP solver the“SoPlex” solver, which has been implemented by Roland Wunderling as apart of his Ph.D. thesis entitled “Paralleler und ObjektorientierterSimplex-Algorithmus” (in German). Information about this solver isavailable at the following website:

http://www.zib.de/Optimization/Software/Soplex/.

[0631] Also, as mentioned above, the task of the LP solver is toidentify an LP solution that minimizes one or more objective functionswhile satisfying a number of constraints. The embodiments describedbelow specify the following objective function for the LP propagation.

minimize: LlnXtYeApB+ . . .+ LLnQtWeDpCeApD+ . . .

[0632] This objective function minimizes the total length. Specifically,each term in this function represents a configuration (i.e., a completeselection of propagations of paths for a net in a child slot), and ismultiplied by the estimated length of that configuration (L1, LL).

[0633] Also, the embodiments described below specify three constraints.The first constraint requires the LP solver to pick only oneconfiguration for every slot-net, as indicated below.

nXtY: nXtYeApB . . . eQpZ+nxtYeApC . . . eQpR+ . . .=1;

[0634] One such constraint is defined for each slot-net across the 16child-slots of the most recently solved slot. This constraint serves tolimit number of selected configurations to 1 per slot-net.

[0635] The second constraint is a propagation consistency constraint,which serves to ensure coherency between child slots (e.g., ifpropagation B is chosen for path A in child slot Y, then the same choicemust be made in the other child slot incident to path A). Thisconstraint can be specified as follows:

nXeYpZ: nXt 0 eYpZeQp 1+nXt 0 YpZeQp 2+nXt 0 eYpZeQp 7 . . . −nXt 1eYpZeSp 1−nXt 1 eYpZeSp 2−nXt 1 eYpZeSp 3=0

[0636] Note that there will be as many positive terms as there areconfigurations specifying propagation B for path A in child slot Y fornet X, and there will be as many negative terms as there areconfigurations specifying propagation B for path A in child slot W fornet X.

[0637] The third constraint is a capacity constraint. Some embodimentsmap the slot-net configurations in the child slots to usage of pathsbetween the grandchild slots (i.e., map each propagation in the childslots to use of paths between the grandchild slots). These embodimentsthen ensure that the capacity of the paths between grandchild slot arerespected.

[0638] The formulation of the LP-propagation problem in some of theseembodiments is as follows:

[0639] [prepPropagationILP(slot)]

[0640] initialize slack=0

[0641] while we don't have a solution

[0642] initialize a new LP solver

[0643] declare the objective function, “objective”

[0644] for each path of the slot

[0645] for each propagation of the path

[0646] retrieve all paths comprising the propagation

[0647] for each path of the propagation

[0648] retrieve the (child-slot,grandchild-slot) pairs that serve asendpoints of the path (a child-slot,grandchild-slot pair may occur inmore than one propagation)

[0649] if this (child-slot,grandchild-slot) pair has not yet beenprocessed

[0650]  create a constraint “tAsBtCsD”, where A is a child-slot, B is agrandchild-slot of child-slot A, C is a child-slot, and D is agrandchild-slot of child-slot C. This constraint will limit the use ofthe path between the grandchild-slots.

[0651] declare a constraint “totlen” to define a total length variable

[0652] for each slot-net X

[0653] for each path Y in the route for the slot-net

[0654] for each propagation Z of that path

[0655] declare a constraint, “nXeYpZ”, to force the LP solver to choosethe same propagation for an identical path in both of its incidentchild-slots

[0656] for each child-slot upon which the route of the current slot-netis incident

[0657] declare a constraint, “nXtY”, where Y is the number of thechild-slot. This is destined to select one configuration per slot-net ineach child-slot

[0658] // finished declaring constraints, now turn to variables

[0659] for each slot-net

[0660] for each child-slot upon which the route of the current slot-netis incident

[0661] identify all configurations of slot-net in child-slot // doneaccording to process 7900

[0662] for each generated configuration

[0663] create variable “nAtBeCpD” where A,B,C,D are integers identifyingthe net, slot, path and propagation, respectively of the config

[0664] for each path in the config

[0665]  if propagation for the path is “unuseable”, add a penalty to theconfig cost // where unuseability determined based on the congestionestimate obtained from the estimates produced by processes 7700 and 7800

[0666]  declare nAtBeCpD to be present in constraint “nAeCpD” withfactor 1.0 if B is the lesser index of the 2 child-slots incident tothis path, −1.0 otherwise

[0667]  for each sub-path in the propagation of the path

[0668]  retrieve the 2 (child-slot, grandchild-slot) pairs that serve asendpoints of the sub-path

[0669]  declare nAtBeCpD to be present in the constraint correspondingto this pair of (child-slot,grandchild-slot)s with factor 0.5

[0670] declare nAtBeCpD to be present in constraint “nAtB” with factor1.0

[0671] declare nAtBeCpD to be present in constraint “totLen”with factorequal to the config cost, stored in the configuration's data structure,plus any penalty

[0672] create variable “tl” to represent the total length of the configsselected

[0673] declare tl to be present in constraint “totLen” with factor −1.0

[0674] declare tl to be present in the objective function with factor1.0

[0675] set the rhs of constraint “totLen”=0.0

[0676] for each path of the slot

[0677] for each propagation of the path

[0678] retrieve all paths comprising the propagation

[0679] for each path of the propagation

[0680] retrieve the (child-slot,grandchild-slot) pairs that serve asendpoints of the path (a child-slot,grandchild-slot pair may occur inmore than one propagation)

[0681] if this (child-slot,grandchild-slot) pair has not yet beenprocessed

[0682]  create a constraint “tAsBtCsD”, where A is a child-slot, B is agrandchild-slot of child-slot A, C is a child-slot, and D is agrandchild-slot of child-slot C. This constraint will limit the use ofthe path between the grandchild-slots.

[0683]  set the rhs of constraint “tAsBtCsD” to the default capacity ofthe propagation-path plus the local variable “slack” value minus the sumof the estimate path use of the propagation and the blocked capacity ofthe propagation path // where the estimated path use was computed byprocess 7800 and the blocked capacity was computed by process 7700

[0684] for each slot-net A

[0685] for each path B in the route of the slot-net

[0686] for each propagation C of that path

[0687] set rhs of constraint “nAeBpC” to 0.0.

[0688] for each child-slot B upon which the route for this slot-net isincident

[0689] set the rhs of constraint “nAtB” equal to 1.0

[0690] solve the LP

[0691] if a solution was found, break out of while loop; otherwise setslack=slack+1 and start again

[0692] As indicated above, the second-to-last line of the formulationtells (at 7620) the LP solver to solve the problem. The LP solver thentries to solve this problem. Each time the LP solver fails to solve thisproblem, the formulation above increments the slack value until the LPsolver is able to solve the problem.

[0693] The LP solver returns a real-number optimal solution. The process7600 then converts this solution to an integer LP (“ILP”) solution. Someembodiments use randomized rounding to perform this conversion, asdescribed above. One of ordinary skill will understand that, instead ofusing an LP solver to generate an LP solution and converting the LPsolution into an ILP solution, other embodiments use an ILP solver forthe propagator to generate an ILP solution.

[0694] e. Follow-Up Propagation

[0695] When the current slot's level is at least two levels above theleaf level, the process 7600 adds the propagation paths that itidentified at 7625 to the follow-up propagation list for the next lowerrecursion level. FIG. 65 illustrates one example of propagation pathsthat can be added to the follow-up propagation list. As mentioned above,this figure illustrates a net that has actual pins 6525 in slots 0 and9. The selected route for this net uses paths P17 and P24 that traversechild slot 5 to connect child slots 0 and 9.

[0696]FIG. 65 illustrates that the path P24 is propagated into childslots 0 and 5 by paths 6510 and 6515, while the path P17 is propagatedinto child slots 5 and 9 by path 6520. The propagation path 6510 isbetween the child slot 7 of the current slot's child slot 0 and thechild slot 8 of the current slot's child slot 1. The propagation path6515 is between the child slot 13 of the current slot's child slot 1 andthe child slot 2 of the current slot's child slot 5. The propagationpath 6520 is between the child slot 14 of the current slot's child slot5 and the child slot 2 of the current slot's child slot 9. FIG. 65illustrates five virtual pins that have been added to the slots of childslots 1, 5, and 9.

[0697] When the current slot's level is at least two levels above theleaf level, the process 7600 adds the propagation paths 6510, 6515, and6520 to the follow-up propagation list for the next lower recursionlevel. The propagator will then use this list when performing follow-uppropagation for a child slot of the current slot. This propagationoperation propagates the paths on the follow-up propagation list onelevel further down.

[0698]FIG. 81 illustrates a process 8100 for performing follow-uppropagation for the current slot when the current slot is below thetop-level slot but above the leaf-level slot. By definition, such acurrent slot is a child slot of a previous parent slot. As shown in FIG.81, the process 8100 initially determines (at 8105) either (1) whetherthe follow-up propagation list includes any path that has at least oneanchor, or (2) whether the current slot is the last slot of the currentlevel and the follow-up propagation list still includes one or morepaths.

[0699] If the process identifies no paths at 8105, the process ends.Otherwise, the process 8100 selects (at 8110) one of the identifiedpaths, and removes this path from the follow-up propagation list. Next,the process costs (at 8115) each propagation permutation of the selectedpath. The cost of each propagation permutation includes the cost of itspropagation path(s) plus the routing cost of the pin configurations inthe two or three child slots that the propagation permutation traverses.

[0700] Next, the process selects (at 8120) the lowest cost propagationpermutation. The selected propagation permutation includes one and insome cases two propagation paths. For instance, in the exampleillustrated in FIG. 65, the propagation of path P24 resulted in twopropagation paths 6510 and 6515, while the propagation of path 17resulted in one propagation path 6520.

[0701] For each propagation path, some embodiments maintain a slot-pairrecord, which stores the identity of the child slots that the path joinsand the remaining capacity of the path. Accordingly, at 8125, theprocess determines whether a slot-pair data structure exists for eachpropagation path that forms the propagation permutation selected at8120. When such a structure does not exist for a propagation path of theselected propagation permutation, the process (at 8125) creates a slotpair structure for the path, stores in the structure the identity of thechild slots that the path traverses, and initializes the capacity fieldof the structure. The initialized capacity for a propagation path is thedefault capacity of the path minus any blockages on the path. When aslot-pair structure already exists for a propagation path of theselected propagation permutation, the process identifies (at 8125) thepath's remaining capacity from this structure.

[0702] At 8130, the process determines whether the selected propagationpermutation can be embedded in the current slot's child slots. In otherwords, the process determines whether the propagation path or paths thatform the selected propagation permutation have a remaining capacitygreater than a threshold value. In some embodiments, the threshold valueis 0. In these embodiments, the selected propagation permutation isembeddable when all the paths that form it have a remaining capacitygreater than zero.

[0703] If the process determines that the selected propagationpermutation cannot be embedded, it determines (at 8135) whether thereare additional propagation permutations for the path selected at 8120.If so, the process selects (at 8140) the next cheapest propagationpermutation and then transitions to 8125.

[0704] When the process determines (at 8135) that there are noadditional propagation permutations to examine, the process embeds (at8160) the best propagation permutation that it examined at 8130. Thisembedding might entail setting virtual pins in the pin distributions ofaffected child slots (i.e., setting virtual pins in the current slot'sgrandchild slots that the selected propagation permutation's path orpaths traverse). One example of setting such virtual pins is illustratedin FIG. 82. This figure illustrates (1) a path 6510 from the follow-uppath list that is propagated into slot 11 of child slot 7 by path 8205,and (2) a virtual pin 8210 that has been set in slot 11 to account forthis propagation. This figure also illustrates that the path 6510 hasbeen propagated along path 8215 into slot 12 of child slot 4 and slot 1of child slot 8 of the slot adjacent to the current slot 8220. Thisfigure also illustrates two virtual pins that have been set in slots 12and 1 of the adjacent slot's child slots 4 and 8.

[0705] At 8160, the process also updates the available capacity of thepropagation path or paths used by the embedded propagation permutation.As mentioned above, the available capacity of a propagation path can becomputed as the default capacity of the path minus the sum of itsblocked capacity and its path use estimate, where the blocked and usevalues are computed according to the processes 7700 and 7800. Someembodiments might not factor the path use estimate computed by usingprocess 7800 in the available capacity of each propagation path.

[0706] When the current slot's level is at least two levels above theleaf level, the process (at 8160) also adds the embedded propagationpath(s) to the follow-up propagation list for the next lower recursionlevel. From 8160, the process transitions to 8150, which will bedescribed below.

[0707] When the process determines (at 8130) that a selected propagationpermutation can be embedded, it embeds (at 8145) the propagationpermutation selected at 8120 or 8140. This embedding might entailsetting virtual pins in the pin distributions of affected child slots(i.e., setting virtual pins in the current slot's grandchild slots thatthe selected propagation permutation's path or paths traverse). At 8145,the process also updates the available capacity of the propagation pathor paths used by the embedded propagation permutation. When the currentslot's level is at least two levels above the leaf level, the process(at 8145) also adds the embedded propagation path(s) to the follow-uppropagation list for the next lower recursion level. From 8145, theprocess transitions to 8150.

[0708] At 8150, the process determines whether it has examined all thepaths that it identified at 8105. If not, the process returns to 8110 toselect another unexamined path that it identified at 8105. If so, theprocess ends.

[0709] 2. Sequential Propagator

[0710] Some embodiments of the invention use a sequential propagationapproach to identify how to propagate the routes specified by the solverinto the current slot's child slots. Some of these embodiments use suchan approach when they use the diagonal propagation model of FIG. 63.

[0711]FIG. 83 illustrates one a sequential-propagation process that isused in some embodiments. As shown in this figure, this process startsby computing (at 8305) the available capacity of each propagationbetween the child slots of the current slot's child slots. The availablecapacity of each propagation path equals the default capacity of thepath minus its blocked capacity plus its path use estimate. As mentionedabove, processes 7700 and 7800 can be used to compute the blocked andpath use values. Some embodiments might not factor the path use estimatecomputed by using process 7800 in the available capacity of eachpropagation path. After computing the available propagation capacities,the process selects (at 8310) a slot-net in the current slot. It thenretrieves (at 8315) the route for the selected slot-net. At 8325, theprocess then selects a path with the most number of anchors. Asmentioned above, some embodiments define an anchor as a pin in eitherchild slot upon which the path is incident. In these embodiments, a pathhas at most two anchors. Other embodiments might define anchor as thenumber of pins in the slots of a child slot; under such an approach, apath can have up to 32 anchors, when it has 16 pins in the 16 slots ofeach child slot.

[0712] Next, the process costs (at 8330) each propagation permutationsof the selected path. The cost of each propagation permutation includesthe cost of the permutation's propagation path(s) plus the routing costof the pin configurations in the two or three child slots that thepropagation permutation traverses.

[0713] Next, the process selects (at 8335) the lowest cost propagationpermutation. The selected propagation permutation includes one and insome cases two propagation paths. For instance, in the exampleillustrated in FIG. 65, the propagation of path P24 resulted in twopropagation paths 6510 and 6515, while the propagation of path 17resulted in one propagation path 6520.

[0714] At 8340, the process determines whether the selected propagationpermutation can be embedded in the current slot's child slots. In otherwords, the process determines whether embedding the selected propagationpermutation will cause any propagation path for this permutation to beover congested.

[0715] If the process determines that the selected propagationpermutation cannot be embedded, it determines (at 8345) whether thereare additional propagation permutations for the path selected at 8325.If so, the process selects (at 8350) the next cheapest propagationpermutation and returns to 8340 to determine whether the newly-selectedpermutation can be embedded.

[0716] When the process determines (at 8345) that there are noadditional propagation permutations to examine, the process embeds (at8365) the best propagation permutation that it encountered at 8340. Thisembedding might entail setting virtual pins in the pin distributions ofaffected child slots (i.e., setting virtual pins in the current slot'sgrandchild slots that the selected propagation permutation's path orpaths traverse). When the current slot's level is at least two levelsabove the leaf level, this embedding also entails adding the propagationpaths used by the selected propagation permutation to the follow-uppropagation list for the next lower recursion level. At 8365, theprocess also updates the available capacity of the propagation paths ofthe embedded propagation permutation. From 8365, the process transitionsto 8360, which will be described below.

[0717] When the process determines (at 8340) that a selected propagationpermutation can be embedded, it embeds (at 8355) the selectedpropagation permutation. This embedding might entail setting virtualpins in the pin distributions of affected child slots (i.e., settingvirtual pins in the current slot's grandchild slots that the selectedpropagation permutation's path or paths traverse). When the currentslot's level is at least two levels above the leaf level, this embeddingalso entails adding the propagation paths used by the selectedpropagation permutation to the follow-up propagation list for the nextlower recursion level. At 8355, the process also updates the availablecapacity of the propagation paths of the embedded propagationpermutation. From 8355, the process transitions to 8360.

[0718] At 8360, the process determines whether it has examined all thepaths of the selected slot-net's route. If not, the process returns to8325 to select another path of this route. If so, the process determines(at 8370) whether it has examined all the slot-nets in the current slot.

[0719] If the process has not examined all the slot-nets in the currentslot, the process returns to 8310 to select another slot-net. Otherwise,the process transitions to 8375. When the current slot's level is afterthe top level but before the leaf level, the sequential propagator thenperforms (at 8375) a follow-up propagation operation. This operationpropagates the routing paths specified by the propagator at the previousrouting level one level further down. When the current slot's level isthe level immediately before the leaf level (i.e., when the current slotis a grandparent of Gcells), the sequential propagator calls (at 8380)the saver to link to the dBNets the path data structures of anypropagation path embedded at 8355, 8365, and 8375. The process thenends.

VII. The Computer System

[0720]FIG. 84 presents a computer system with which one embodiment ofthe present invention is implemented. Computer system 8400 includes abus 8405, a processor 8410, a system memory 8415, a read-only memory8420, a permanent storage device 8425, input devices 8430, and outputdevices 8435.

[0721] The bus 8405 collectively represents all system, peripheral, andchipset buses that communicatively connect the numerous internal devicesof the computer system 8400. For instance, the bus 8405 communicativelyconnects the processor 8410 with the read-only memory 8420, the systemmemory 8415, and the permanent storage device 8425.

[0722] From these various memory units, the processor 8410 retrievesinstructions to execute and data to process in order to execute theprocesses of the invention. The read-only-memory (ROM) 8420 storesstatic data and instructions that are needed by the processor 8410 andother modules of the computer system. The permanent storage device 8425,on the other hand, is read-and-write memory device. This device is anon-volatile memory unit that stores instruction and data even when thecomputer system 8400 is off. Some embodiments of the invention use amass-storage device (such as a magnetic or optical disk and itscorresponding disk drive) as the permanent storage device 8425. Otherembodiments use a removable storage device (such as a floppy disk orzip® disk, and its corresponding disk drive) as the permanent storagedevice.

[0723] Like the permanent storage device 8425, the system memory 8415 isa read-and-write memory device. However, unlike storage device 8425, thesystem memory is a volatile read-and-write memory, such as a randomaccess memory. The system memory stores some of the instructions anddata that the processor needs at runtime. In some embodiments, theinvention's processes are stored in the system memory 8415, thepermanent storage device 8425, and/or the read-only memory 8420.

[0724] The bus 105 also connects to the input and output devices 8430and 8435. The input devices enable the user to communicate informationand select commands to the computer system. The input devices 8430include alphanumeric keyboards and cursor-controllers.

[0725] The output devices 8435 display images generated by the computersystem. For instance, these devices display IC design layouts. Theoutput devices include printers and display devices, such as cathode raytubes (CRT) or liquid crystal displays (LCD).

[0726] Finally, as shown in FIG. 84, bus 8405 also couples computer 8400to a network 8465 through a network adapter (not shown). In this manner,the computer can be a part of a network of computers (such as a localarea network (“LAN”), a wide area network (“WAN”), or an Intranet) or anetwork of networks (such as the Internet).

[0727] Any or all of the components of computer system 8400 may be usedin conjunction with the invention. However, one of ordinary skill in theart would appreciate that any other system configuration may also beused in conjunction with the present invention.

[0728] While the invention has been described with reference to numerousspecific details, one of ordinary skill in the art will recognize thatthe invention can be embodied in other specific forms without departingfrom the spirit of the invention. For instance, several embodiments weredescribed above by reference to a hierarchical router, one of ordinaryskill will realize that other embodiments of the invention areimplemented with other router types, such as maze routers.

[0729] Also, even though several embodiments were described by referenceto an LP-problem formulation, one of ordinary skill will realize thatthese embodiments can be practiced by applications that do not utilizean LP solver. The above-described track sharing constraints provide onesuch example. Any type of router can account for these sharingconstraints in determining whether to embed routes.

[0730] In addition, other embodiments might use different set ofpartitioning lines to divide the circuit layout. For example, someembodiments might use partitioning grids that define different-shapedand/or different-sized sub-regions than the sub-regions defined by the4×4 grid illustrated in FIG. 5. Thus, one of ordinary skill in the artwould understand that the invention is not to be limited by theforegoing illustrative details, but rather is to be defined by theappended claims.

We claim:
 1. A hierarchical method of routing nets within a particularregion of a circuit layout, each net having a set of pins, the methodcomprising: a) partitioning the particular region into a first set ofsub-regions; b) for each net, identifying a first route that connects agroup of first-set sub-regions containing the first net's pins; whereinsome of the routes have at least one route-edge that is at leastpartially diagonal; c) partitioning the sub-regions into a second set ofsmaller sub-regions; d) for a first net, identifying a propagation ofthe first-net's first route into the second-set sub-regions based oncongestion between the second-set sub-regions; e) adjusting thecongestion between the second set sub-regions based on the identifiedpropagation; f) for a second net, identifying a propagation of thesecond-net's first route into the second-set sub-regions based oncongestion between the second-set sub-regions.
 2. The method of claim 1,wherein a first set of paths exists between the first set ofsub-regions, and a second set of paths exists between the second set ofsub-regions, wherein each net's first route traverses a sub-set of thefirst set of paths.
 3. The method of claim 2, wherein identifying thepropagation of the first-net's first route comprises: for eachparticular first-set path traversed by the first-net's first route,identifying a second-set path that represents the propagation of theparticular first set path.
 4. The method of claim 3, wherein at leastone diagonal second set path represents the propagation of at least onediagonal first-set path traversed by the first-net's first route.
 5. Themethod of claim 3, wherein at least one diagonal second set pathrepresents the propagation of at least one Manhattan first-set pathtraversed by the first-net's first route.
 6. The method of claim 3further comprising for at least one of the particular first-set pathstraversed by the first-net's first route, identifying two second-setpath that together represent the propagation of the particular first-setpath into the second set of sub-regions.
 7. The method of claim 6,wherein the at least one particular first-set path is diagonal, and thetwo identified second-set paths are diagonal.
 8. The method of claim 3,wherein adjusting the congestion comprises adjusting a set of congestionvalues, wherein each congestion value specifies the congestion along asecond-set path.
 9. The method of claim 1, wherein a first set ofinter-region edges exists between the first set of sub-regions, and asecond set of inter-region edges exists between the second set ofsub-regions, wherein each net's first route intersects a sub-set of thefirst set of inter-region edges.
 10. The method of claim 9, whereinidentifying the propagation of the first-net's first route comprises:for each particular first-set edge intersected by the first-net's firstroute, identifying a second-set edge that represents the propagation ofthe particular first set edge.
 11. The method of claim 9, wherein atleast one diagonal second-set edge represents the propagation of atleast one diagonal first-set edge intersected by the first-net's firstroute.
 12. The method of claim 9, wherein at least one diagonalsecond-set edge represents the propagation of at least one Manhattanfirst-set edge intersected by the first-net's first route.
 13. Themethod of claim 9 further comprising for at least one of the particularfirst-set edges intersected by the first-net's first route, identifyingtwo second-set edges that together represent the propagation of theparticular first-set edge into the second set of sub-regions.
 14. Themethod of claim 13, wherein the at least one particular first-setinter-region edge is diagonal, and the two identified second-setinter-region edges are diagonal.
 15. The method of claim 9, whereinadjusting the congestion comprises adjusting a set of congestion values,wherein each congestion value specifies the congestion along asecond-set path.
 16. The method of claim 1 wherein the pins are actualand virtual pins for the net.
 17. The method of claim 1, wherein avirtual pin in one of the first set of sub-regions is set to account forpropagation of a higher-level route into the particular region.
 18. Themethod of claim 1 further comprising for the first net defining at leastone virtual pin in a second-set sub-region to account for propagatingthe first-net's first route into the second set of sub-regions.